Helicity/Chirality Helicities of (ultra-relativistic) massless - - PowerPoint PPT Presentation

Helicity/Chirality

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• Helicities of (ultra-relativistic) massless

particles are (approximately) conserved

• Conservation of chiral charge is a property of

massless Dirac theory (classically)

• The symmetry is anomalous at quantum level

Right-handed Left-handed

Chiral magnetic effect

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• Chiral charge is produced by topological

QCD configurations

• Random fluctuations with nonzero chirality

in each event

• Driving electric current



 

a a f L R

F F x d N g dt N N d

 

 ~ 16 ) (

3 2 2

฀ NR  NL  0  5  0

5 2 2

2   B e j   

Heavy ion collisions

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• Dipole pattern of electric currents (charge

correlations) 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)]

Experimental evidence

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

Chiral separation effect

• Axial current induced by fermion chemical

potential (free theory!)

[Vilenkin, Phys. Rev. D 22 (1980) 3067] [Metlitski & Zhitnitsky, Phys. Rev. D 72, 045011 (2005)] [Newman & Son, Phys. Rev. D 73 (2006) 045006]

• Exact result (is it?), which follows from

chiral anomaly relation

• No radiative correction expected…

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฀ j5

3 free   eB

2 2 

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?

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Possible implication

• Seed chemical potential (μ) induces axial

current

• Leading to separation of chiral charges:

μ5>0 (one side) & μ5<0 (another side)

• In turn, chiral charges induce back-to-back

electric currents through

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฀ j5

3 free   eB

2 2  ฀ j 3

free   e2B

2 2 5

Quadrupole CME

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• 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]

Motivation

• Any additional consequences of the CSE

relation? (free theory!)

[Metlitski & Zhitnitsky, Phys. Rev. D 72, 045011 (2005)]

• Any dynamical parameter ∆ (“chiral shift”)

associated with this condensate?

• Note: ∆=0 is not protected by any symmetry

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฀ j5

3 free   eB

2 2 

   

5 3

   L L

Chiral shift in NJL model

[Gorbar, V.M., Shovkovy, Phys. Rev. C 80, 032801(R) (2009)]

• NJL model (local interaction)
• “Gap” equations:

(“effective” chemical potential) (dynamical mass) (chiral shift parameter)

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฀   0  1 2 Gint j 0 m  m0  Gint      1 2 Gint j5

3

Solutions

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

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Chiral shift @ Fermi surface

• Chirality is ≈ well defined at Fermi surface
• L-handed Fermi surface:
• R-handed Fermi surface:

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฀ n  0 : k 3   (  s

)2  m2

n  0 : k 3   ( 2  2n eB  s

)2  m2

k 3   ( 2  2neB  s

)2  m2

฀ n  0 : k 3   (  s

)2  m2

n  0 : k 3   ( 2  2n eB  s

)2  m2

k 3   ( 2  2n eB  s

)2  m2

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 the

conventional axial anomaly relation

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Axial current

• Does the chiral shift give any contribution to

the axial current?

• In the point splitting method, one has

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

• This is consistent with the NJL calculations
• Since , the correction to the

axial current should be finite

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฀ j5

 singular  

 2 2 2 

3  2

2 2 

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Axial current in QED

[Gorbar, V.M., Shovkovy, Wang, Phys. Rev. D 88, 025025 (2013);

• ibid. D 88, 025043 (2013)]
• Lagrangian density
• Axial current
• To leading order in coupling α=e2/(4π)

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 

) , ( tr

5 3 2 5 3

x x G Z j    

฀ L   1 4 F F  i  D   0  m

 

 (counterterms)



   ) , ( ) , ( ) , ( ) , ( ) , (

4 4

y v S v u u x S v d u d i y x S y x G

Expansion in external field

• Use expansion of S(x,y) in powers of
• To leading order in coupling,
• The radiative correction is

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฀ j5

3 0 

฀ A

ext

฀ j5

3  

฀ A

ext

฀ A

ext

฀ A

ext

฀ A

ext

Alternative form of expansion

• Expand in field
• The Schwinger phase (in Landau gauge)
• Note: the phase is not translation invariant

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฀ S(x,y)  S

(0)(x  y) S (1)(x  y)i(x,y)S(0)(x  y)

Translation invariant part Schwinger phase

฀ (x,y) eB 2 (x1  y1)(x2  y2) ฀ S(x,y)  ei(x,y)S (x  y)

Translation invariant parts

• Fourier transforms
• Note the singularity near the Fermi surface…

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฀ S

(0)(k)  i

(k0  ) 0 k   m k0   i sign(k0)

 

2 k2  m2

 

 

2 2 2 3 3 2 1 ) 1 (

) ( sign ) ( ) ( m k i k m k k eB k S         

2

k       

Fermi surface singularity

• “Vacuum” + “matter” parts

where

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 

 

" Mat. " " Vac. " ) ( sign 1

2 2

     

n

m k i k

2

k  

  

  

 

2 2 ) 1 ( 1

• )!

1

• (

) 1 (- 2 = " Mat. " m k k k n i

n n

    

 2

k       

 

 

n

i m k      

2 2

1 = " Vac. "

2

k

Axial current (0th order)

• From definition
• After integrating over energy

and finally

• Note the role of the Fermi surface (!)

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฀ j5

3 0  

d4k 2

 

4 tr  3 5S (1)(k)

 



฀ j5

3 0  eBsign()

4 3 d3k 2 k2  m2

 



฀ j5

3 0  eBsign()

2 2 2  m2

Matter part

Conventional wisdom

• Only the lowest (n=0) Landau level contributes

giving same answer

• There are no contributions from higher Landau

levels (n≥1)

• There is a connection with the index theorem

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฀ j5

3 0  eB

4 2 d k3    k3

2  m2

 

   k3

2  m2

 

     



฀ j5

3 0  eBsign()

2 2 2  m2

Two facets

• Two ways to look at the same result

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฀ B 0 ฀ B  0

Radiative correction

• Original two-loop expression
• After integration by parts

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Result (m<<μ)

• Loop contribution
• Counterterm
• Final result

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฀ j5

3 ct  eB 

2 3 ln  m ln m

2

m2  9 4       eBm2 2 3 ln  m  3 4       ฀ f1  f2  f3  eB 2 3 ln  2  11 12       eBm2 2 3 ln  23/2  1 6       ฀ j5

3   eB

2 3 ln 2 m ln m

2

m2  4 3       eB m2 2 3 ln 23/2 m  11 12      

• Unphysical dependence on photon mass
• Infrared physics with

not captured properly

• Note: similar problem exists in calculation of

Lamb shift

Sign of nonperturbative physics

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฀ m  k0, k3  eB

฀ j5

3   eB

2 3 ln 2 m ln m

2

m2  4 3       eB m2 2 3 ln 23/2 m  11 12      

Nonperturbative effects (?)

• Perpendicular momenta cannot be defined

with accuracy better than (In contrast to the tacit assumption in using expansion in powers of B-field)

• Screening effects provide a natural infrared

regulator (Formally, this goes beyond the leading order in coupling)

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฀ k

 min ~

eB ฀ m  

Nonperturbative result (?)

• Conjectured nonpertubative modification

(1) If non-conservation of momentum dominates (2) If photon screening is more important

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฀ j5

3   eB

2 3 ln  eB m3 O 1

 

       eBm2 2 3 ln  eB O 1

 

        ฀ j5

3   eB

2 3 ln  3 m3 O 1

 

      eB m2 2 3 ln 1  O 1

 

     

Self-energy at B≠0

• Self-energy
• General structure
• Translation invariant part:

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฀ (x,y)  4i  S(x,y) D (x  y) ฀ (x,y)  exp i(x,y)

 

(x  y) ฀  (p)  4i d4k 2

 

4   S

(k) D (k  p)



Contribution linear in B

• The result has the form

where

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฀ 

(1)(p)  4i

d4k 2

 

4   S (1)(k) D (k  p)



฀ 

(1)(p)   3 5  0 55(p)

฀    eB   m2 ln m2 2 p  pF

 

1         ฀ 5(p)    eB   m2 p3 pF ln m2 2 p  pF

 

1        

Dispersion relations

• Let us use the condition

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฀ Det iS

1(p)   (1)(p)

  0

L/R-Fermi surface shift

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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 in

studying a new class of materials, Weyl semimetals, in condensed matter.

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