Transcript CPODD 2012, BNL Some Field Theoretical Issues of the Chiral Magnetic Effect Hai-cang Ren The Rockefeller University & CCNU with De-fu Hou, Hui Liu JHEP 05(2011)046

CPODD 2012, BNL
Some Field Theoretical Issues of
the Chiral Magnetic Effect
Hai-cang Ren
The Rockefeller University & CCNU
with De-fu Hou, Hui Liu
JHEP 05(2011)046
1
The contents
•
•
•
•
•
•
An introduction of CME
Axial anomaly in QCD
General properties of CME
One loop calculation
Summary
Current project
2
I. An introduction to CME
(Fukushima, Kharzeev and Warringa)
1. A charged massless quark in a magnetic field B 
Helicity   5
L
charge
+
—
+
—
Magnetic moment




Momentum








Current J
B
R
JR
JL
In a quark matter of net axial charge
Q5
e2
J  JR  JL 
5B
2
2
  NC  q2f  Color-flavor factor
f
3
2. RHIC Implementation
5  0
i) Excess axial change
Transition between different topologies of QCD
T=0
Axial anomaly N5  
N f g2
32 2
T≠0
4
l
l
d
x

F
F
     nW
nW  the wind number Fl  QCD field strength
B
ii) Magnetic field
Generated by an off-central collision
ion

ion

4
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
potentials
local equilibrium
＊Beyond thermal equilibrium
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3. The robustness of
2
J CME
e

5B
2
2
under the Infrared limit of
 5 k , k 0 

1
1 

J q  k，  k 0  
2
2 

i.e.
(k, k0 )  0
k0 
1

 B q  k,   
2
2

and
(q, )  0
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A relativistic quantum field theory at nonzero temperature
and/or chemical potentials
• UV divergence is no worse than vacuum
• IR is more problematic because:
p0
---- The appearance of the ratio
|p|
limits p0  0 and p  0 may dependon orders
|p|
---- The appearance of the ratios
etc.
T
T  0 and p  0 may dependon orders
---- Linde’s problem with gluons.
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II. Axial anomaly in QCD
• Naïve Ward identities:
 J  0
  J 5  0
and
J   i   qˆ  EM current
J 5   i    5  axial current
qˆ  charge matrixin flavorspace
• 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:
 PV with mass M PV and negativemetricin Hilbert space
  J PV  0
but
  J 5PV  2iM PV PV  5 PV
Fermionloop of  
Fermionloop of    C PV  fermionloop of  PV
 CPV  1
PV
PV
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• Ward identities post regularization:
M PV  
 J  0
N f g2
2
e
l
l
  J 5  i

F
F
  F F
    i
2
2
32
16
anomalousWard identity
Fl  YM field strength F  EM field strenth
  N c  q 2f  color - flavorfactor
f
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• Applications of the axial anomaly:
0
※ The explanation of the rate of   2
※ 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.
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III General properties of CME
i) Naïve axial charge & conserved axial charge
5   5
†
N 5   d 3 r 5
dN5
e2
e2
3  5
3
3
3
 d r
   d r  J 5  
d
rE

B


d
rE  B
2 
2 
dt
t
4
4
not conservedbecause of theanomaly
e2
~
3
N5  N5 
d
rA  B
2 
4
~
dN 5
~
 0  N 5 is conserved!
dt
~
N 5 should be used in thermodynamics equilibrium (Rubakov)
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ii) Grand partition function:
~
 H  N   5 N 5 

Z  T rexp 

T


N  quark number,  ,  5  chemicalpotentials
iii) Linear response
Ji (Q)  Kij (Q) Aj (Q)
Q  (q, )
e2
Kij (Q)   ij (Q)  i 2 5 ijk qk  O( A2 )
4
 ij (Q)  The usual photon self-energy tensor, subject to
higher order corrections
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iv) The Taylor expansion in 5
Kij (Q)  Kij(0) (Q)  5 Kij(1) (Q)  O(52 )
Chiral magnetic current J CME
2

e
Kij(1) (Q)  
ij (Q)
 i 2  ijk qk


0
5
5
2
K 
Q1 

 i 5 
 Q2

Normal term
K
Anomaly term
 i 5 
1
1 

Q1   q  k ,   k 0 
2
2 

1
1 

Q2    q  k ,  k 0 
2
2 

K  (k, ik 0 )
  lim
e2
 i
 q
2 ijk k
2
K
Q1 

K 0
 Q2
Q1 

 i 5 
 Q2


  (Q1 , Q2 )
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The limit K  (0, k 0 )  0
1 

Q1   q,   k 0 
2 

1 

Q2    q,  k 0 
2 

K 
Q1 

 i 5 
 Q2

K
 i 5 
 Q2
Q1 


1
e2
lim lim  ij (Q1 , Q2 )  i lim K   ij (Q1 , Q2 )  i 2  ijk qk
k0 0 k 0
k0 0 k
2
0

anomalousWard identity
K
Q1 

J CME  0
to all orders and all T and
 i 5 
 Q2


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The limit K  (k,0)  0
K
1


Q1  q1 ,     q  k ,  
2


1


Q2  q 2 ,     q  k , 
2


Q1 

 i 5 
 Q2

General tensor structure with Bose symmetry:
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The electromagnetic gauge invariance:
2
e
Q1  Q2  (q, )
K (Q)  i 2 5  F (Q)  1  ijk qk
2
F (Q)  C 0 (q 2 , q 2 ,q 2 ;  )  C 0 (q 2 , q 2 ,q 2 ; )
(1)
ij


; )
 q 2 C1 (q 2 , q 2 ,q 2 ;  )  C1 (q 2 , q 2 ,q 2 ; )

  C 2 ( q 2 , q 2 , q 2 ;  )  C 2 ( q 2 , q 2 , q 2
F (Q)  0
If the infrared limit exists: lim
Q 0
e2
K (Q)  i 2 5 ijk qk
2
(1)
ij
J CME
e2

5B
2
2
to all orders
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III. One loop calculation
3
d
p 

K ij(1) (Q)  T  

(
P
,
Q
|
0
)

C

(
P
,
Q
|
M
)

PV ij
PV 
3  ij
(
2

)
p0
PV


 ij ( P, Q | m)  trS ( P  Q | m) i S ( P | m) j
S ( P | m) 
C
PV
i
 i  P   4   5 4 5  m
P  (p, p0 ), Q  (q,  )
1
PV
Continuation of imaginary Matsubara q0 to   i0  with  real
for retarded (advanced) response function after the summation
over Matsubara p0
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Subtlety of IR limit:
e2
K (Q)  i 2 5  F (Q)  1  ijk qk
2
(1)
ij
lim lim F (Q)  0
T 0
and / or
0
q 0  0
2
lim lim F (Q ) 
 0 q 0
3
J CME
e2

5B
2
2
J CME
e2
  2 5B
6
Kharzeev
& Warringar
IR singularity:
C 2 (0,0,0;  ) 
T 0
and
 0
1
3
lim lim F (Q)  lim lim F (Q)  1
q 0  0
 0 q 0
J CME  0
IR singularity:
C1 (q 2 , q 2 ;q 2 ;  ) 
1
2 q2   2


and C 2 (q 2 , q 2 ;q 2 ;  )  


2 q2   2

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III. Summary
1
1 

J CME  q  k ,   k 0 
2
2 

1
1 

vs.  5 k , k 0  & B q  k ,   k 0 
2
2 

J CME
IR limit
T 0
 0
Higher order
lim lim
k0 0 k 0
0
none
lim lim
k 0 k0 0
J CME
T 0
 0
T 0
and / or
 0
lim lim
k0 0 k 0
lim
lim
k 0 k0 0
lim lim
k 0 k0 0
lim lim
limlim
q 00   0
limlim
  q0

2
5 B
2e 2
1 2
 5 B
3 2e 2
k0 0 k 0
lim lim
T 0
and/or
q0  0
lim lim
k 0 k0 0
lim lim
 0 q0
0
lim lim
e2
 2 5B
2
1
e2

5B
2
3
2
0
none if IR safe
yes
none
k0 0 k 0
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Current project
Anomalous transport coefficients
T heLagrangiandensity
       iA  5    5  5  ...
AnomalousWardidentity:
1
i
  J 5  
  F F
2
3 48
Anomaly& thermodyn
amic laws dictates Son & Surowka



J 5   B B  


B  axial magneticfield
  fluid vorticity
One loop without regulators Landsteiner et. al.
B 
1
2
2
5
2
T
  2  52 
2
2
1
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One loop with P V regulators:
Using theconservedaxial chargedictatedby anomaly:
1
1
rm)
  5    5   2 A  B(anomaly te
3 4
 B ,    P V regulatedone loop  anomaly term
Regulated one loop
B

1
3
1
2 2
2
5
1
6
 52  T 2
Anomaly term
1
6
2
0
5
Total
1
2
1
2 2
2
5
1
6
 52  T 2
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Anomaly and thermodynamics
AnomalousWardidentityand energy - momentumconservation
  J 5  E  B 
  T   F  J 5
and
  anomalycoefficient
u   T    5   J 5  F  J 5  E  B 
But we found for P V regulators

  J 5PV
  TPV 



u   TPV 
 2M PV PV  5 PV but
 F   J 5 PV   2 A PV  5 PV


  5   J 5PV


 F   J 5 PV 
Does the anomaly still show up in the regulated

u  T


 5   J 5
?
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Thank you!
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