Helicity/Chirality• Helicities of (ultra-relativistic) masslessparticles are (approximately) conserved
Right-handed
Left-handed• Conservation of chiral charge is a property ofmassless Dirac theory (classically)• The symmetry is anomalous at quantum level2

Chiral magnetic effect• Chiral charge is produced by topological
QCD configurationsd (NR  NL ) g N f3
 ~ ad xFa F2 dt162• Random fluctuations with nonzero chirality
in each event
NR  NL  0  5  0• Driving electric current2e Bj  2 52
3

Heavy ion collisions• Dipole pattern of electric currents (chargecorrelations) in heavy ion collisions[Kharzeev, Zhitnitsky, Nucl. Phys. A 797, 67 (2007)][Kharzeev, McLerran, Warringa, Nucl. Phys. A 803, 227 (2008)][Fukushima, Kharzeev, Warringa, Phys. Rev. D 78, 074033 (2008)]
4

Experimental evidence[B. I. Abelev et al. [The STAR Collaboration], arXiv:0909.1739][B. I. Abelev et al. [STAR Collaboration], arXiv:0909.1717]
5

Chiral separation effect• Axial current induced by fermion chemicalpotentialj

3
5 freeeB 22(free theory!)[Vilenkin, Phys. Rev. D 22 (1980) 3067][Metlitski & Zhitnitsky, Phys. Rev. D 72, 045011 (2005)]
 & Son, Phys. Rev. D 73 (2006) 045006][Newman• Exact result (is it?), which follows fromchiral anomaly relation• No radiative correction expected…
6

The chiral anomaly and CSE
Ambjorn, Greensite, Peterson (1983): Only LLL generates the chiral anomaly.

Axial current induced in CSE:
In a free theory,is generated only in LLL.

The connection between

and:Then,(anomalous relation!)Is the relation

exact?

7

Possible implication• Seed chemical potential (μ) induces axialcurrentj

35 freeeB 22

• Leading to separation of chiral charges:
μ5>0 (one side)&

μ5<0 (another side)• In turn, chiral charges induce back-to-backelectric currents through2eB
3
j free   2 528

Quadrupole CME• Start from a small baryon density and B≠0• Produce back-to-back electric currents[Gorbar, V.M., Shovkovy, Phys. Rev. D 83, 085003 (2011)][Burnier, Kharzeev, Liao, Yee, Phys. Rev. Lett. 107 (2011) 052303]
9

Motivation• Any additional consequences of the CSE
relation?
j

35 freeeB 22

(free theory!)[Metlitski & Zhitnitsky, Phys. Rev. D 72, 045011 (2005)]• Any dynamical parameter ∆ (“chiral shift”)
associatedwith this condensate?

L  L0     
35• Note: ∆=0 is not protected by any symmetry
10Chiral shift in NJL model[Gorbar, V.M., Shovkovy, Phys. Rev. C 80, 032801(R) (2009)]• NJL model (local interaction)• “Gap” equations:10  0  Gint j2m  m 0  Gint 1   Gint j 53
2(“effective” chemical potential)(dynamical mass)(chiral shift parameter)
11

Solutions• Magnetic catalysis solution (vacuum state):• State with a chiral shift (nonzero density):

12

Chiral shift @ Fermi surface• Chirality is ≈ well defined at Fermi surface• L-handed Fermi surface:n 0:

k 3   (  s ) 2  m 2n 0:

k 3   (  2  2n eB  s ) 2  m 2k 3   (  2  2n eB  s ) 2  m 2• R-handed Fermi surface:n 0:

k 3   (  s ) 2  m 2n 0:

k 3   (  2  2n eB  s ) 2  m 2k 3   (  2  2n eB  s ) 2  m 213

Chiral shift vs. axial anomaly• Does the chiral shift modify the axial
anomaly relation?
• Using point splitting method, one derives[Gorbar, V.M., Shovkovy, Phys. Lett. B 695 (2011) 354]

• Therefore, the chiral shift does not affect theconventional axial anomaly relation
14

Axial current• Does the chiral shift give any contribution tothe axial current?
• In the point splitting method, one has 3j5 singular   2 2   2 2 2


2

3

[Gorbar, V.M., Shovkovy, Phys. Lett. B 695 (2011) 354]

• This is consistent with the NJL calculations• Since, the correction to theaxial current should be finite
15

Axial current in QED[Gorbar, V.M., Shovkovy, Wang, Phys. Rev. D 88, 025025 (2013);
ibid. D 88, 025043 (2013)]• Lagrangian density
1 L   F F   i   D   0  m   (counterterms)
4• Axial current
53j  Z 2 tr   G ( x, x)
35

• To leading order in coupling α=e2/(4π)
G ( x, y )  S ( x, y )  i  d ud vS ( x, u )(u , v)S (v, y )
4

4

16

Expansion in external field• Use expansion of S(x,y) in powers of A ext
• To leading order in coupling,extj35 0

A• The radiative correction
is
j3
5 extextAextAA

17

Alternative form of expansion• Expand S(x, y)  e i(x,y)S (x  y) in fieldS(x, y)  S (x  y)  S (x  y)  i(x, y) S (x  y)
(0)

(1)

Translation invariant part(0)

Schwinger phase• The Schwinger phase (in Landau gauge)eB(x, y)   (x1  y1)(x 2  y 2 )2• Note: the phase is not translation invariant18Translation invariant parts• Fourier transforms
S (k)  i(0)(k0  )  k   m
022k
i
sign(k)
km
00 

S (1) (k )   1 2 eB

2(k0   ) 0  k3 3  m

k    i sign(k )  k  m 2022 20• Note the singularity near the Fermi surface…
19

Fermi surface singularity• “Vacuum” + “matter” parts1k    i sign(k )  k  m 202 n2

" Vac."" Mat."0where" Vac." =1k     k  m  i 222n0

2 i(-1) n-1" Mat." =    k0   k0   ( n 1) k0   2  k 2  m 2(n - 1)!
20Axial current (0th order)• From definition4j3
5 0d k3 5 (1)tr S (k)4
2 • After integrating over energy

j3
5 0eB sign( )
3222
dk

k
m



34

and finally
j

3
5 0

Matter parteB sign( )22m2 2• Note the role of the Fermi surface (!)
21

Conventional wisdom• Only the lowest (n=0) Landau level contributesj35 0 eB
2222 
   k3  m     k3  m 2  d k 3 4giving same answer
j

3
5 0

eB sign( )
22

m
2 2

• There are no contributions from higher Landaulevels (n≥1)
• There is a connection with the index theorem
22Two facets• Two ways to look at the same result
B0B 023

Radiative correction• Original two-loop expression• After integration by parts

24

Result (m<<μ)• Loop contribution
 eB    11  eB m 2  1 f1  f 2  f 3  ln 3 / 2  3 ln32  2  12  2   2  6 

• Counterterm
22 

m
eB

9
eBm
3
j 53 ct  ln
ln
 ln
32 32  mm
4  2   m 4 

• Final result
22 3/2 2 meB
4eBm2
11
j 53    ln 2   ln3 ln32  mm
3  2   m12 25

Sign of nonperturbative physics• Unphysical dependence on photon mass
22 
3/2

 2 m
eB
4
eB
m
2
11

j 53   ln
ln
 ln32 
32  mm
3  2   m12 • Infrared physics withm  k0 , k3  eB

not captured properly• Note:similarproblemexistsincalculationof
Lamb shift26

Nonperturbative effects (?)• Perpendicular momenta cannot be definedwith accuracy better thank min ~eB(In contrast to the tacit assumption in using
expansion in powers of B-field)• Screening effects provide a natural infrared
regulatorm  

(Formally, this goes beyond the leading orderin coupling)
27

Nonperturbative result (?)• Conjectured nonpertubative modification(1) If non-conservation of momentum dominates
2 
eB
eB
eBm

j 53   lnO13 ln
3  O132  m 2  eB(2) If photon screening is more important
32 eBeBm
1
3j5   
 O1ln
3 ln3  O132  m 2  
28

Self-energy at B≠0• Self-energy(x, y)  4i   S(x, y)  D (x  y)

• General structure(x, y)  exp i(x, y)( x  y)

• Translation invariant part:4d k (p)4i
S(k)
D (k  p)
 2 429

Contribution linear in B
4d k  (1) ( p)  4i  4  S (k)  D (k  p)
2 (1)• The result has the form (p)        5 ( p)(1)where3505

 eB  m2ln12 
 m  2 p  pF  

2

 eB  p3m

5 ( p)  ln
12 m pF 2pp F  30

Dispersion relations• Let us use the condition

Det i S ( p)   ( p)  01(1)31

L/R-Fermi surface shift32Summary• Radiative corrections in CSE are nonzero. New
face of the chiral anomaly.
• Chiral shift
is generated in magnetized matter.
It induces a chiral asymmetry on the Fermi
surface and contributes to the axial current.
• Radiative corrections vanish without “matter”
part with singularities on Fermi surface.
• In 2011, the chiral shift
was rediscovered instudying a new class of materials, Weyl
semimetals, in condensed matter.
33