Helicity/Chirality. Helicities of (ultra-relativistic) massless particles are (approximately) conserved Right-handed

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2 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

3 Chiral magnetic effect Chiral charge is produced by topological QCD configurations d 2 ( N R N L ) f d 2 dt g Random fluctuations with nonzero chirality in each event N R N L N 16 xf a ~ F a Driving electric current e 2 B j 2 2 5

4 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 80, 227 (2008)] [Fukushima, Kharzeev, Warringa, Phys. Rev. D 78, 0740 (2008)] 4

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

6 Chiral separation effect Axial current induced by fermion chemical potential j 5 free eb 2 2 (free theory!) [Vilenkin, Phys. Rev. D 22 (1980) 067] [Metlitski & Zhitnitsky, Phys. Rev. D 72, (2005)] [Newman & Son, Phys. Rev. D 7 (2006) ] Exact result (is it?), which follows from chiral anomaly relation No radiative correction expected 6

7 The chiral anomaly and CSE Ambjorn, Greensite, Peterson (198): 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

8 Possible implication Seed chemical potential (μ) induces axial current Leading to separation of chiral charges: μ 5 >0 (one side) & μ 5 <0 (another side) j 5 In turn, chiral charges induce back-to-back electric currents through j free eb 2 2 free e2 B

9 Quadrupole CME Start from a small baryon density and B 0 Produce back-to-back electric currents [Gorbar, V.M., Shovkovy, Phys. Rev. D 8, (2011)] [Burnier, Kharzeev, Liao, Yee, Phys. Rev. Lett. 107 (2011) 0520] 9

10 Motivation Any additional consequences of the CSE relation? (free theory!) [Metlitski & Zhitnitsky, Phys. Rev. D 72, (2005)] Any dynamical parameter ( chiral shift ) j 5 free eb 2 2 associated with this condensate? L L 0 Note: =0 is not protected by any symmetry 5 10

11 Chiral shift in NJL model [Gorbar, V.M., Shovkovy, Phys. Rev. C 80, 02801(R) (2009)] NJL model (local interaction) Gap equations: G int j 0 m m 0 G int 1 2 G int j 5 ( effective chemical potential) (dynamical mass) (chiral shift parameter) 11

12 Solutions Magnetic catalysis solution (vacuum state): State with a chiral shift (nonzero density): 12

13 Chiral Fermi surface Chirality is well defined at Fermi surface L-handed Fermi surface: n 0 : k ( s ) 2 m 2 n 0 : k ( 2 2n eb s ) 2 m 2 k ( 2 2n eb s ) 2 m 2 R-handed Fermi surface: n 0 : k ( s ) 2 m 2 n 0 : k ( 2 2n eb s ) 2 m 2 k ( 2 2n eb s ) 2 m 2 1

14 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) 54] Therefore, the chiral shift does not affect the conventional axial anomaly relation 14

15 Axial current Does the chiral shift give any contribution to the axial current? In the point splitting method, one has j 5 singular [Gorbar, V.M., Shovkovy, Phys. Lett. B 695 (2011) 54] This is consistent with the NJL calculations Since, the correction to the axial current should be finite 15

16 Axial current in QED [Gorbar, V.M., Shovkovy, Wang, Phys. Rev. D 88, (201); ibid. D 88, (201)] Lagrangian density L 1 4 F F i D 0 m (counterterms) Axial current j 5 Z tr 5 G( x, x) 2 To leading order in coupling α=e 2 /(4π) G( x, y) S( x, y) i d 4 ud 4 vs( x, u) ( u, v) S( v, y) 16

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

18 Alternative form of expansion Expand S(x,y) e i(x,y) S (x y) in field S(x,y) S (0) (x y) S (1) (x y) i(x,y)s (0) (x y) Translation invariant part Schwinger phase 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

19 Translation invariant parts Fourier transforms S (0) (k) i (k 0 ) 0 k m k 0 i sign(k 0 ) 2 k 2 m 2 S (1) ( k) 1 2 eb k isign( k ) k m 2 0 ( k 0 ) 0 0 k m Note the singularity near the Fermi surface 19

20 Fermi surface singularity Vacuum + matter parts k 0 1 isign( k 0 ) 2 k 2 m 2 n "Vac." "Mat." where "Vac."= k k 2 m 2 i n n-1 2 i(-1) ( n1) 2 2 " Mat."= k0 k0 k0 k m ( n -1)! 2 20

21 Axial current (0 th order) From definition j 5 d 4 k 0 2 tr 4 5 S (1) (k) After integrating over energy j 5 ebsign() d k 2 k 2 m and finally Matter part j 5 0 ebsign() m 2 Note the role of the Fermi surface (!) 21

22 Conventional wisdom Only the lowest (n=0) Landau level contributes j 5 eb d k k 2 m 2 k 2 m 2 giving same answer There are no contributions from higher Landau j 5 levels (n 1) 0 ebsign() m 2 There is a connection with the index theorem 22

23 Two facets Two ways to look at the same result B 0 B 0 2

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

25 Loop contribution Result (m<<μ) f 1 f 2 f eb ln eb m2 2 ln 2 / Counterterm j 5 ct eb ln 2 m ln m 2 m eb m2 2 ln m 4 Final result j 5 eb ln 2 2 m ln m 2 m 4 2 eb m2 2 ln 2 / 2 11 m 12 25

26 Sign of nonperturbative physics Unphysical dependence on photon mass j 5 eb ln 2 2 m ln m 2 m 4 2 eb m2 2 ln 2 / 2 11 m 12 Infrared physics with m k 0, k eb not captured properly Note: similar problem exists in calculation of Lamb shift 26

27 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) k ~ eb min m 27

28 Nonperturbative result (?) Conjectured nonpertubative modification (1) If non-conservation of momentum dominates j 5 eb ln eb 2 O 1 m eb m2 2 ln eb O 1 (2) If photon screening is more important j 5 eb ln O1 2 m eb m2 2 ln 1 O 1 28

29 Self-energy General structure Self-energy at B 0 (x,y) 4i S(x,y) D (x y) Translation invariant part: (p) 4i (x,y) exp i(x,y) (x y) d 4 k 2 4 S (k) D (k p) 29

30 Contribution linear in B (1) (p) 4i d 4 k 2 4 S (1) (k) D (k p) The result has the form where (1) (p) (p) eb m 2 ln 5 (p) eb m 2 p p F m 2 2p p F ln m 2 1 2p p F 1 0

31 Dispersion relations Let us use the condition DetiS 1 (p) (1) (p) 0 1

32 L/R-Fermi surface shift 2

33 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.