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Strong magnetic fields in
lattice QCD
P.V.Buividovich, M.N.Chernodub ,T.K. Kalaydzhyan,
D.E. Kharzeev, E.V.Luschevskaya , M.I. Polikarpov
arXiv:1003.2180, arXiv:0910.4682, arXiv:0909.2350,arXiv:0909.1808, arXiv:0907.0494, arXiv:0906.0488,
arXiv:0812.1740
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Lattice simulations with magnetic fields
1. Chiral Magnetic Effect1.1 CME on the lattice
1.2 Vacuum conductivity induced by magnetic field
1.3 Quark mass dependence of CME
1.4 Dilepton emission rate
2. Other effects induced by magnetic field2.1 Chiral symmetry breaking2.1 Chiral symmetry breaking
2.2 Magnetization of the vacuum
2.3 Electric dipole moment of quark along the direction of the magnetic field
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Magnetic fields in non-central collisions[Fukushima, Kharzeev, Warringa, McLerran ’07-’08]
Main picture by McLerran,Kharzeev,Fukushima
Heavy ion
Heavy ion
Quarks and gluons
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Magnetic fields in non-central collisions[Fukushima, Kharzeev, Warringa, McLerran ’07-’08]
Main picture by McLerran,Kharzeev,Fukushima
[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.
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Magnetic fields in non-central collisions
Charge is largeVelosity is high
Thus we have
Main picture by McLerran,Kharzeev,Fukushima
The medium is filled by electrically charged particles
Large orbital momentum, perpendicular to the reaction plane
Large magnetic field along the direction of the orbital momentum
Thus we havetwo very bigcurrents
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Magnetic fields in non-central collisions
Two very bigcurrentsproduce a verybig magneticfield
B
“Instanton”
McLerran,Kharzeev,Fukushima
The medium is filled by electrically charged particles
Large orbital momentum, perpendicular to the reaction plane
Large magnetic field along the direction of the orbital momentum
field
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2eB Λ≈
In heavy ion collisionsmagnetic forces are of the order of
strong interaction forces
2
QCDeB Λ≈
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Magnetic forces are of the order of strong interaction forces
2
QCDeB Λ≈
We expect the influence of magnetic field on strong interaction physics
QCDeB Λ≈
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Chiral Magnetic Effect by Fukushima, Kharzeev, Warringa, McLerran
1. Massless quarks in external magnetic field.
Red: momentum Blue: spin
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Chiral Magnetic Effect by Fukushima, Kharzeev, Warringa, McLerran
1. Massless quarks in external magnetic field.
Red: momentum Blue: spin
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Chiral Magnetic Effect by Fukushima, Kharzeev, Warringa, McLerran
2. Quarks in the instatnton field.Red: momentumBlue: spin
Effect of topology:Effect of topology:
uL → uR
dL → dR
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Chiral Magnetic Effect by Fukushima, Kharzeev, Warringa, McLerran
3. Electric current along magnetic fieldRed: momentumBlue: spin
Effect of topology:Effect of topology:
uL → uR
dL → dR
u-quark: q=+2/3d-quark: q= - 1/3
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Chiral Magnetic Effect by Fukushima, Kharzeev, Warringa,
McLerran3. Electric current is along
magnetic fieldIn the instanton field
Red: momentumRed: momentumBlue: spin
Effect of topology:uL → uRdL → dR
u-quark: q=+2/3d-quark: q= - 1/3
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3D time slices of topological charge density,
lattice calculations
D. Leinweber
Topological charge density after
vacuum cooling
P.V.Buividovich,
T.K. Kalaydzhyan, M.I. Polikarpov
Fractal topological charge density
without vacuum cooling
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Magnetic forces are of the order of strong interaction forces
We expect the influence of magnetic field on
2
QCDeB Λ≈We expect the influence of magnetic field on
strong interaction physics
The effects are nonperturbative,
and we use
Lattice Calculations
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fieldmagneticexternalHr
We calculate µνµ σγψψ ,1, ; =Γ>Γ<
in the external magnetic field and in thepresence of the vacuum gluon fields Weconsider SU(2) gauge fields andquenched approximation
1
0
2
3
T
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Quenched vacuum, overlap Dirac operator, external magnetic field
2 qkπ 2
2)250(;
2MeveB
L
qkeB ≥=
π
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Density of the electric charge vs. magnetic field, 3D time slices
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1. Chiral Magnetic Effect on the lattice, numerical results T=0
ψγψ 33
2
3
2
3
2
3 ,)0,0(),( =><−>=<>< jjTHjj IR
Regularized electric current:
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Chiral Magnetic Effect on the lattice, numerical comparison of resultsnearTc and nearzero
>>=<<
>>=<<
≠
=
22
2
2
2
1
12 0
0
jj
jj
F
T
>>≠<<
>>=<<
≠
>
22
2
2
2
1
12 0
0
jj
jj
F
T
ψγψ iiiiIRi jjTHjj =><−>=<>< ,)0,0(),( 222
>>=<< 2
0
2
3 jj
Regularized electric current:
>>≠<< 2
0
2
3 jj
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Chiral Magnetic Effect, EXPERIMENT VS LATTICE DATA (Au+Au)
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Chiral Magnetic Effect, EXPERIMENT VS LATTICE DATA
fmR 5≈
experiment
fm
fm
fmR
1
2.0
5
≈
≈
≈
τ
ρ
our fit
D. E. Kharzeev,L. D. McLerran, and H. J. Warringa, Nucl. Phys. A 803, 227 (2008),
our lattice data at T=350 Mev
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1.2 Magnetic Field Induced Conductivity of the Vacuum
Qualitative definition of conductivity, s
yxmACyjxj
−−⋅+=><
|}|exp{)()(
C
xqxqxj
r
yxmACyjxj
∝
=
−−⋅+=><
σ
γ µµ
ανµ
)()()(
|}|exp{)()(
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Magnetic Field Induced Conductivity of the Vacuum
- Conductivity (Kubo formula)
Maximal entropy method
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Magnetic Field Induced Conductivity of the Vacuum
- Conductivity (Kubo formula)
For weak constant electric fieldFor weak constant electric field
kiki Ej σ>=<
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Magnetic Field Induced Conductivity of
the Vacuum
Calculations in SU(2) gluodynamics
We use overlap operator + Shifted Unitary Minimal Residue Method
(Borici and Allcoci (2006)) to obtain fermion propagator
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Magnetic Field Induced Conductivity of the Vacuum
Calculations in SU(2) gluodynamics
T = 0 T/Tc = 1.12
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Calculations in SU(2) gluodynamics,
conductivity along magnetic field at
T=0
axis Z0
toparallel isHv
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Calculations in SU(2) gluodynamics,
conductivity along magnetic field at
T=0
At T=0, B=0 vacuum is insulator
Critical value of magnetic field?
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Calculations in SU(2) gluodynamics,
conductivity along magnetic field at
T=0, T>0
axis Z0
toparallel isHv
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Calculations in SU(2) gluodynamics,
conductivity at T=0, variation of the
quark mass
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Calculations in SU(2) gluodynamics,
conductivity at T=0, variation of the
quark mass and magnetic field
ji
ijmB
BB∝σ
Why?Why?Why?Why?
q
ijmB
∝σ
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1.3 Dilepton emission rateL. D. McLerran and T. Toimela, Phys. Rev. D 31, 545 (1985),
E. L. Bratkovskaya, O. V. Teryaev, and V. D.Toneev, Phys. Lett. B 348, 283 (1995)
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• There should be more soft dileptons in the
direction of magnetic field
BpBpd ))((rrrr⋅⋅σ
qmB
BpBp
dpdp
d
||
))(( 21
21
⋅⋅∝
σ
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2. Other effects induced by magnetic field
2.1 Chiral symmetry breaking2.2 Magnetization of the vacuum 2.3 Electric dipole moment of quark along the 2.3 Electric dipole moment of quark along the direction of the magnetic field
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3. Chiral condensate in QCD
><−=Σ ψψ
><= ψψ22 ><= ψψππ qmfm 22
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Chiral condensate vs. field strength, SU(2) gluodynamics
We are in agreement with the chiral perturbationtheory: the chiral condensate is a linear function ofthe strength of the magnetic field!
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Chiral condensate vs. field strength, SU(3) gluodynamics
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�� � �� � �� � � � � � �� � � � �� �� � � � � � � �� �� � � � � �� � �� � � �� � � �� � � � �� �� � �� � � � � � � �� � � � � � � � �� � �� � � � � � �� � � �� �� � � � � � �� � � �� �� � � � �� � � �� �� �� �� � � � � � �� � �� � � � � � � � � � � � � � � � � � � �� � �� � � � � � � � � � � �� � � � � � � �� � � � �� �� � � � �� �� � � � � � � � � � � � �� � �� � �
Localization of Dirac Eigenmodes
Typical densities of the nearzero eigenmodes vs. the strength of the external magnetic field
B=0B=0
B=(780Mev)2
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4. Magnetization of the vacuum as a function of the magnetic field
Spins of virtual quarks turn parallel to the magnetic field
2003.) Ball, P. 1985, Balitsky, I. (I.
rules sum QCD50
resultour )3(46
↔−≈><
↔−=><
Mev
Mev
χψψ
χψψ
],[2
1βααβ
αβαβ
γγσ
ψψχψσψ
i
F
=
><>=<
2003.) Ball, P. 1985, Balitsky, I. (I.
rules sum QCD50
resultour )3(46
↔−≈><
↔−=><
Mev
Mev
χψψ
χψψ
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5. Generationof the anomalous quark electric dipole moment
along the axis of magnetic field
ψγψρ 55 =yyyyLarge correlation between square of the electric dipole moment
and chirality ψγγψσ ],[ 00 ii i=
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6. Electric currents in
instanton field+magnetic field (CME)
ψγψρ 55 =The fluctuations of the chirality
and the fluctuations of the longitudinal
electric current as a function of the
magnetic field.
ψγψ iiiiIRi jjTHjj =><−>=<>< ,)0,0(),( 222
The squared components of the
electric current in a 12-plane. The
upper sheet represents the spatial
distribution of the longitudinal
current, the lower sheet
corresponds to the transverse
current.
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• 1. We obsreve signatures of the Chiral Magnetic Effect, butthe physics may differ from the model of Kharzeev, McLerranand Warringa (arXiv:0907.0494, Phys.Rev.D79:106003,2009)
• 2. We observe that in the confinement phase the externalmagnetic field induces nonzero electric conductivity along the
Conclusions
magnetic field induces nonzero electric conductivity along thedirection of the magnetic field, transforming the system froman insulator into an anisotropic conductor. In thedeconfinement phase the conductivity does not exhibit anysizable dependence on the magnetic field (arXiv:1003.2180).
• 3. The conductivity is weaker for heavy quarks, thus it isinteresting to measure experimentally the charge asymmetryfor S and C quarks.
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• 4. 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, Phys.Lett. B 682:484-489,2010 ).
Conclusions
(arXiv:0812.1740, Phys.Lett. B 682:484-489,2010 ).
• 5. The calculated vacuum magnetization is in a qualitative
agreement with model calculations (arXiv:0906.0488,
Nucl.Phys. B 826 (2010) 313).
• 6. We observe very large correlation between electric dipole
moment of quark and chirality (arXiv:0909.2350
Phys.Rev.D81:036007,2010).
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Systematic errors
• SU(2) gluodynamics instead of QCD
• Moderate lattice volumes
• Not large number of gauge field configurations
• In some cases we calculate the overlap
propagator using summation over
eigenfunctions: