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

Helicity/Chirality • Helicities of (ultra-relativistic) massless particles are (approximately) conserved Right-handed Left-handed • Conservation of chiral charge is a property of massless Dirac theory (classically) • The symmetry is anomalous at quantum level 2

Chiral magnetic effect • Chiral charge is produced by topological QCD configurations  2 g N ( ) d N N ~    f 3 a R L d x F F   a 2 16 dt • Random fluctuations with nonzero chirality in each event N R  N L  0   5  0 • Driving electric current   2 e B   j ฀  5 2 2 3

Heavy ion collisions • 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)] 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 chemical potential free   eB 2  2  3 j 5 (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… 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 axial current free   eB 2  2  3 j 5 • Leading to separation of chiral charges: μ 5 >0 (one side) & μ 5 <0 (another side) ฀ • In turn, chiral charges induce back-to-back electric currents through free   e 2 B 2  2  5 j 3 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? free   eB 2  2  3 j 5 (free theory!) [Metlitski & Zhitnitsky, Phys. Rev. D 72 , 045011 (2005)] • Any dynamical parameter ∆ (“chiral shift”) ฀ associated with this condensate?  L       3 5 L 0 • 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:    0  1 2 G int j 0 (“effective” chemical potential) m  m 0  G int   (dynamical mass)    1 3 2 G int j 5 (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: k 3   (   s   ) 2  m 2 n  0 : k 3   (  2  2 n eB  s   ) 2  m 2 n  0 : k 3   (  2  2 neB  s   ) 2  m 2 • R-handed Fermi surface: k 3   (   s   ) 2  m 2 n  0 : ฀ k 3   (  2  2 n eB  s   ) 2  m 2 n  0 : k 3   (  2  2 n eB  s   ) 2  m 2 13 ฀

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 14

Axial current • Does the chiral shift give any contribution to the axial current? • In the point splitting method, one has  3   2   singular   2  2  2   2  2   3 j 5 [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 15

Axial current in QED [Gorbar, V.M., Shovkovy, Wang, Phys. Rev. D 88 , 025025 (2013); ibid. D 88, 025043 (2013)] • Lagrangian density L   1   4 F  F    i   D    0  m   (counterterms) • Axial current   ฀     5 3 5 j Z tr G ( x , x ) 3 2 • To leading order in coupling α = e 2 /(4π)     4 4 G ( x , y ) S ( x , y ) i d u d v S ( x , u ) ( u , v ) S ( v , y ) 16

Expansion in external field • Use expansion of S ( x,y ) in powers of ext A  • To leading order in coupling, ext A  0  3 j 5 ฀ • The radiative correction is ฀ ฀ ext ext ext A  A  A    3 j 5 ฀ ฀ ฀ 17 ฀

Alternative form of expansion S ( x , y )  e i  ( x , y ) S • Expand in field ( x  y ) S ( x , y )  S (0) ( x  y )  S (1) ( x  y )  i  ( x , y ) S (0) ( x  y ) ฀ Schwinger phase Translation invariant part ฀ • The Schwinger phase (in Landau gauge)  ( x , y )   eB 2 ( x 1  y 1 )( x 2  y 2 ) • Note: the phase is not translation invariant 18 ฀

Translation invariant parts • Fourier transforms ( k 0   )  0  k    m (0) ( k )  i S 2  k 2  m 2   k 0    i  sign( k 0 )       0 3 ( k ) k m     ฀ ( 1 ) 1 2   0 3 ( ) S k eB   2       2 2 k 2 k i sign ( k ) m 0 0 • Note the singularity near the Fermi surface… 19

Fermi surface singularity • “Vacuum” + “matter” parts 1     " Vac. " " Mat. "   n       2 2 k 2 k i sign ( k ) m 0 0 where 1   " Vac. " =   n       2 2 k 2 k m i 0       n - 1 2 i (- 1 )                2 2 k ( n 1 ) 2 " Mat. " = k k k m 0 0 0 ( n - 1 )! 20

Axial current (0 th order) • From definition d 4 k    0   4 tr  3  5 S 3 (1) ( k ) j 5   2  • After integrating over energy 0   eB sign(  ) d 3 k   2  k 2  m 2    ฀ 3 j 5 4  3 and finally Matter part 0   eB sign(  )  2  m 2 3 ฀ j 5 2  2 • Note the role of the Fermi surface (!) 21 ฀

Conventional wisdom • Only the lowest (n=0) Landau level contributes       0  eB 2  m 2 2  m 2  d k 3         3 j 5 k 3 k 3     4  2 giving same answer 0   eB sign(  ) ฀  2  m 2 3 j 5 2  2 • There are no contributions from higher Landau levels (n≥1) ฀ • There is a connection with the index theorem 22

Two facets • Two ways to look at the same result B  0 B  0 ฀ ฀ 23

Radiative correction • Original two-loop expression • After integration by parts 24

Result (m<< μ ) • Loop contribution     f 1  f 2  f 3   eB  ln    eBm 2  2   11 2 3/2   1    ln 2  3 2  3   12   6  • Counterterm     ฀ 2 ct    eB  ln    eBm 2 ln  m  ln m  m 2  9  3   3  j 5 2  3 2  3  4 m  4     • Final result         eB  ln 2  2   eB m 2 ln 2 3/2  m  ln m  m 2  4  11 ฀   3  j 5 2  3 2  3  3 m  12     25 ฀

Sign of nonperturbative physics • Unphysical dependence on photon mass         eB  ln 2  2   eB m 2 ln 2 3/2  m  ln m  m 2  4  11   3  j 5 2  3 2  3  3 m  12     • Infrared physics with m   k 0 , k 3  eB ฀ not captured properly • Note: similar problem exists in calculation of ฀ Lamb shift 26

Nonperturbative effects (?) • Perpendicular momenta cannot be defined with 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 regulator m    (Formally, this goes beyond the leading order in coupling) ฀ 27