1

Some Field Theoretical Issues of the Chiral Magnetic Effect

Hai-cang RenThe Rockefeller University & CCNU

with De-fu Hou, Hui Liu

JHEP 05(2011)046

CPODD 2012, BNL

2

The contents

• An introduction of CME

• Axial anomaly in QCD

• General properties of CME

• One loop calculation

• Summary

• Current project

3

(Fukushima, Kharzeev and Warringa)

1. A charged massless quark in a magnetic field

Helicity R L

charge + — + —Magnetic moment

Momentum

Current J

5

I. An introduction to CME

In a quark matter of net axial charge 5Q

2C f

f

N q

B��������������

RJ��������������

LJ��������������

Color-flavor factor

B ��������������

BJJJ 52

2

2

e

LR

4

the wind number QCD field strength

ii) Magnetic field Generated by an off-central collision

2. RHIC Implementationi) Excess axial change

Transition between different topologies of QCD

Axial anomaly

5 0

T=0

T≠0

24

5 232f l l

W

N gN d x F F n

Wn lF

ion

ion

B��������������

5

iii) May provide a new signal of QCD phase transition.

iv) Theoretical approach:

---- Field theory (Fukushima et. al., Kharzeev et. al.)

---- Holographic theory (Yee, Rebhan et. al.)

v) There are experimental evidences, remains to be solidified.

vi) Complication in RHIC: ＊ Inhomogeneous & time dependent magnetic field ＊ Inhomogeneous temperature and chemical poten

tials local equilibrium＊ Beyond thermal equilibrium

6

02

1

2

1k，kqJ

2,

2

1 0kkqB

, 05 kk

3. The robustness of

BJ 52

2

CME 2

e

under the Infrared limit of

i.e. 0),( and 0),( 0 qk k

7

A relativistic quantum field theory at nonzero temperatureand/or chemical potentials

• UV divergence is no worse than vacuum

• IR is more problematic because:

---- The appearance of the ratio

---- The appearance of the ratios etc.

---- Linde’s problem with gluons.

||0

p

p

orderson dependmay 0 and 0 limits 0 pp

T

|| p

orderson dependmay 0 and 0 pT

8

• Naïve Ward identities:

spaceflavor in matrix chargeˆ

current axial current EMˆ

0 and 0

55

5

q

iJqiJ

JJ

II. Axial anomaly in QCD

• UV divergence demands regularization (e.g. Pauli Villars) ------ Not all Ward identities can be preserved ------ The ones related to gauge symmetries have to be maintained

PV regulators:

of loopfermion of loopFermion

of loopFermion

2 but 0

spaceHilbert in metric negative and mass with

PVPVPV

PV5PVPVPV5

PV

PVPV

C

iMJJ

M

PV

PV 1C

9

• Ward identities post regularization:

factorflavor -color

strenth field EMstrength field YM

identity Wardanomalous

1632

0

2

2

2

2

2

5

PV

f

fc

l

llf

qN

FF

FFe

iFFgN

iJ

J

M

10

※ The explanation of the rate of

※ The solution of UA(1) problem

※ Link the change of the axial charge and the change of topology.

※ Chiral magnetic effect, chiral vortical effect, etc.

0 2

• Applications of the axial anomaly:

11

III General properties of CME

conserved! is ~

0~

4

~

55

32

2

55

Ndt

Nd

de

NN

BrA

53

5 rdN

anomaly theof because conservednot 44

32

23

2

2

5335

BrEBrEJrr 5 de

de

dt

ddt

dN

i) Naïve axial charge & conserved axial charge

should be used in thermodynamics equilibrium (Rubakov)

†5 5

5

~N

12

( ) ( ) ( )i ij jJ Q K Q A Q ( , )Q q

22

52( ) ( ) ( )

4( )

ij ij ijk k

ij

eK Q Q i q O A

Q

The usual photon self-energy tensor, subject tohigher order corrections

potentials chemical, number,quark

~expTr

5

55

N

T

NNHZ

ii) Grand partition function:

iii) Linear response

13

(0) (1) 25 5( ) ( ) ( ) ( )ij ij ijK Q K Q K Q O

5

2(1)

205

( ) ( )2ij ij ijk k

eK Q Q i q

Chiral magnetic current

5iv) The Taylor expansion in

CMEJ��������������

Normal term Anomaly term

5i

1Q

2Q

K

0lim

K

5i

1Q

2Q

K

02

01

2

1,

2

1

2

1,

2

1

kQ

kQ

kq

kq

),( 0ikK k

kijkqe

i

2

2

2

5i

1Q 2Q

K

),( 21 QQ

14

5i

1Q

2Q

K

to all orders and all T and0CMEJ

02

01

2

1,

2

1,

kQ

kQ

q

q

kijkijk

ijk

qe

iQQKk

iQQ

2

2

210

021

00 2),(

1lim),(limlim

00

k

The limit 0),0( 0 kK

5i

1Q

2Q

K identity Wardanomalous

5i

1Q 2Q

K

15

The limit 0)0,( kK

General tensor structure with Bose symmetry:

,2

1,

,2

1,

22

11

kqq

kqq

Q

Q

5i

1Q 2Q

K

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The electromagnetic gauge invariance:

2

(1)52

( ) ( ) 12ij ijk k

eK Q i F Q q

0lim ( ) 0Q

F Q

BJ 52

2

2

e

CME

If the infrared limit exists:2

(1)52

( )2ij ijk k

eK Q i q

to all orders

),(21 q QQ

);,,();,,(

);,,();,,(

);,,();,,()(

2222

2222

2221

2221

2

2220

2220

qqqCqqqC

qqqCqqqCq

qqqCqqqCQF

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III. One loop calculation

1

),( ),,( )|(

)|()|(tr)|,(

)|,()0|,()2(

)(

PVPV

05454

PVPVPV3

3)1(

0

C

QpPmPi

imPS

mPSmQPSmQP

MQPCQPd

TQK

jiij

ijijp

ij

qp

p

0qContinuation of imaginary Matsubara to with real for retarded (advanced) response function after the summation over Matsubara

0i

0p

18

2

(1)52

( ) ( ) 12ij ijk k

eK Q i F Q q

0

/

0

T

and or

BJ 52

2

2

e

CME

Subtlety of IR limit:

IR singularity:

IR singularity:

0

0

T

and

Kharzeev& Warringar

BJ 52

2

6

e

CME

0)(limlim00

QFq

3

2)(limlim

00

QF

q

0CMEJ

3

1);0,0,0(2 C

1)(limlim)(limlim0000

QFQFqq

2);;,( and

2

1);;,(

22222

222222

1

qqqqC

qqqqC

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III. Summary

IR limit Higher order

0 none

none if IR safe

yes

0 none

00lim limq

0 0lim limq 0 00l i m l i mkk

0 0 0lim limk k

2 522e B

2 5213 2

e B

0 00lim limkk 0 0 0lim limk k

0

0

T

0

/

0

T

and or

CMEJ

0

0

T

0 00lim lim

kk

0 0 0lim limk k

0 00lim lim

kk

0 0 0lim limk k

00lim limq

0 0lim lim

q

0050 2

1,

2

1 & , vs.

2

1,

2

1kkkCME kqBkkqJ

0

and/or

0

T

CMEJ

B52

2

2

e

B52

2

23

1

e

20

22

1

2

1

regulators without loop One

vorticityfluid field magnetic axial

dictates laws amic thermodyn&Anomaly 483

1

:identity WardAnomalous

...

density Lagrangian The

225252

5

25

555

T

B

BJ

FFi

J

iA

B

B

Son & Surowka

Landsteiner et. al.

Current project

Anomalous transport coefficients

21

rmanomaly te loop one regulated PV,

rm)anomaly te(4

1

3

1

:anomalyby dictated charge axial conserved theUsing

:regulators PV with loop One

255

B

BA

Regulated one loop Anomaly term Total

0

B

523

1

526

1

2252 6

1

2

1T

22

52 6

1

2

1T

522

1

22

2

but 2

regulators PVfor found But we

tcoefficienanomaly

and

onconservati momentum-energy andidentity WardAnomalous

PV55PV5PV

PV5PVPV5PV

PV5PVPV5PV

555

55

JFJTu

AJFT

MJ

BEJFJTu

JFTBEJ

Anomaly and thermodynamics

Does the anomaly still show up in the regulated

55 JTu ?

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Thank you!