Chiral Magnetic Effect Kenji Fukushima (Yukawa Institute for Theoretical Physics) July 24 2010@Yukawa 1
Strong q Angle, Strong CP Problem and Heavy-Ion Collisions July 24 2010@Yukawa 2
P and CP Violation in the YM Theory Gauge Actions □ P - and CP - even ( T -even) terms = 2 F 0 i F 0 i F ij F ij F F Even w.r.t. spatial and temporal indices □ P - and CP - odd ( T -odd) terms = 2 F 01 F F 23 2 F 02 F 31 2 F 03 F 12 F Odd w.r.t. spatial and temporal indices Parallel E and B = 2 E ⋅ B E B F F vector axial vector July 24 2010@Yukawa 3
Terminology Topological Charge (Pontryagin Index) 1 4 x F a 2 ∫ d a Q = F 32 a = 1 a 2 F F 1 a a =∂ K 2 F F 32 2 A c 1 a ∂ A a 1 abc A a A b A K = 3 16 July 24 2010@Yukawa 4
Terminology Chern-Simons Number ijk A i c 1 a 1 3 x abc A i a A j b A k = ∫ d 2 ∫ d 3 x K 0 = a ∂ j A k 3 16 4 x ∂ 0 K 0 −∂ i K i = ∫ dt d 3 x K 0 = t =∞− t =−∞ Q = ∫ d dt ∫ d July 24 2010@Yukawa 5
q -Vacuum and Strong CP Problem Topological Structure and q -Vacuum Manton Faddeev Jackiw-Rebbi ∣ 〉 = ∑ i ∣ 〉 e (Bloch state) Strong CP Problem Spin 〈 ∣ 〉 S QCD = − 1 1 2 tr F 2 tr F F F EDM 16 2 g + 2 ∣ d n ∣~ e m q / m N - No CP breaking (Why?) − 11 ∣∣ 0.7 × 10 July 24 2010@Yukawa 6
Finite- q Hadronic World q can be eliminated by U(1) A rotation One solution to the strong CP problem is the presence of massless quarks (almost excluded...) Effect of strong q -angle to hadron physics i U U e Scalar meson i 5 ~ cos 0 sin ~ e Pseudo-scalar meson i 5 i 5 ~ 0 cos sin 0 ~ i 5 e h 0 condensates in addition to the chiral s condensate July 24 2010@Yukawa 7
Possibility for Finite h Condensate If U(1) A symmetry is NOT broken s and h are degenerate ( h may have a chance as much as the s condensate develops) U(1) A is broken but can be “effectively” restored U(1) A breaking effective interaction is induced by the topological susceptibility Veneziano-Di Vecchia Susceptibility drops off at high temperature T ~ T c 2 exp [ − ] n T = 2 2 N c N f 2 8 2 / g − 5 e − 8 2 2 T 2 3 g Gross-Pisarski-Yaffe Lattice Simulation Alles et al (1996) July 24 2010@Yukawa 8
Relativistic Heavy-Ion Collisions Kharzeev Finite- q Pisarski Tytgat Voloshin Baym Heavy-Ion (nucleus) Quark-Gluon Plasma Shuryak Au, Pb, Cu, ... Direct photon measurement s NN = 200GeV , 62GeV , ... 0 , h ' etc) (not from p 12 K ~ GeV → Initial T ~ 4×10 c.f. T c ~ QCD ~ 0.3fm July 24 2010@Yukawa 9
Topological Contents in the QCD Vacuum and the Real-time Fluctuations July 24 2010@Yukawa 10
Lattice Simulation Topological Charge Distribution at T =0 This is not a function of “Real-Time” but of the simulation step. Derek's Visual QCD July 24 2010@Yukawa 11
Is the high-T QCD Vacuum Topologically Trivial? Yes … in terms of Instantons (Euclidean) Instantons are exponentially suppressed at high T . 2 exp [ − ] n T = 2 2 N c N f 2 8 2 / g − 5 e − 8 2 2 T 2 3 g No … in terms of Sphalerons (Minkowskian) Sphalerons are parametrically enhanced at high T . 5 T 4 ~ s QCD sphalerons are abundant in hot and dense matter created in the relativistic heavy-ion collisions Arnold-McLerran (1987) July 24 2010@Yukawa 12
Topological Rate in Real- and Imaginary-Time Pendulum (Arnold-McLerran) Chern-Simons number x t / 2 Topological charge n = 1 2 ∫ 0 d ˙ x d x 1 2 i Finite- T Euclidean Action S E = ∫ 0 x 2 ˙ 2 ˙ Topological Susceptibility (Diffusion Rate) A t = 〈 T 〉 2 x t − x 0 2 2 2 Real-time (classical approx.) A t ≃ t t 2 = v 2 2 4 4 2 〉≃ 2exp − 2 2 / cos Imaginary-time A − i =〈 n July 24 2010@Yukawa 13
Analytical Continuation Diffusion Rate at High T A t = 〈 T 〉 O e 2 x t − x 0 2 / 2 2 2 m exp m − 1 O e exp − i m ∣ t ∣− 1 exp − m − 1 − exp i m ∣ t ∣− 1 1 2 / 2 = 4 2 i t O e 2 1 t 2 / 2 4 2 / 2 A t =− i O e Instantons (Euclidean windings) are suppressed at high T but communications in real time are not and dominated by the contribution from the zero-winding sector. July 24 2010@Yukawa 14
Topological Diffusion Rate = 1 4 x 〈 q x q 0 q 0 q x 〉 V ∞ ∫ d 2 lim t ∞ lim 2 〉= 2 V t 〈 Q Random Walk at Finite T In the strong-coupling AdS/CFT by Son and Starinets (hep-th/0205051) 2 2 = g YM N 4 3 T 256 In the weak-coupling perturbation by Arnold, Son, Yaffe, Bodeker, Moore, etc 1 2 5 ln 4 = const ⋅ g YM N T 2 g YM N July 24 2010@Yukawa 15
Connection to the Heavy-Ion Collisions How to detect the topological effects? July 24 2010@Yukawa 16
Non-Central Collision Before Collision (seen from above) + + Centrality is determined by N part July 24 2010@Yukawa 17
Non-Central Collision After Collision + B + (Local) P and CP Violation July 24 2010@Yukawa 18
Estimated Magnetic Fields Classical (Pancake) Calcs (Kharzeev-McLerran-Warringa) 2 ] eB = 1 [ MeV 14 [ Gauss ] B ≃ 1.7 × 10 UrQMD Calculations (Skokov-Illarionev-Toneev) July 24 2010@Yukawa 19
How Big? 2 → 10 18 Gauss eB ~ m p 3 ~ 10 6 × 10 Neutron Star (Magnetar) July 24 2010@Yukawa 20
Chiral Magnetic Effect Classical Picture Left-handed Quark = momentum parallel to spin B Right-handed Quark = momentum parallel to spin J ≠ 0 if N 5 = N R − N L ≠ 0 Kharzeev-McLerran-Warringa (2007) Kharzeev-McLerran-Warringa July 24 2010@Yukawa 21
Anomaly Relations Induced N 5 by Topological Effects 2 N f dN 5 = − g 3 x tr F 2 ∫ d F QCD Anomaly Relation dt 8 Introduce m 5 to describe induced N 5 Induced J by the presence of N 5 and B 2 5 j = e 2 B QED Anomaly Relation 2 j = ∑ 2 B in QCD 2 5 q i 2 i = flavor Metlitski-Zhitnitsky (2005) Fukushima-Kharzeev-Warringa (2008) July 24 2010@Yukawa 22
Derivation (naïve calculation) Thermodynamic Potential ( UV divergent ) f ∫ ∣ q f B ∣ dp 3 ∞ 2 [ n , s 2 T ln 1 e − n ,s ] =− V N c ∑ 2 ∑ s =± ∑ n, s f n = 0 2 = p 3 2 2 2 ∣ q f B ∣ n sgn p 3 s 5 2 n, s m Current ( UV finite ) Only surface terms! j 3 = e ∣ eB ∣ 2 ∑ n , s [ n, s p 3 =− n , s p 3 =− ] 4 s, n 2 B 5 n, s s 5 = e = e ∣ eB ∣ 2 ∑ 2 2 2 s ,n July 24 2010@Yukawa 23
Derivation (energy conservation) Energy Conservation (Nielsen-Ninomiya 1983) Electric field E → Energy shift (Fermi energy) Landau Levels Density of states Energy cost E B N N R L July 24 2010@Yukawa 24
CME from Inhomogeneous q Space-time Dependent q -angle i 5 / 2 N f ⋅∂ i ⋅∂/ 2 N f 5 i 5 / 2 N f = e ⋅∂ ⋅∂ e ∂ 0 / 2 N f = 5 Schematic Picture CME ≠ 0 in-medium No CME 〈〉≠ 0 〈 0 〉≠ 0 Kharzeev, Pisarski, = 0 vacuum Tytgat, Krasnitz, 〈〉≠ 0 〈 0 〉= 0 Venugopalan, Voloshin, ... No CME July 24 2010@Yukawa 25
Witten Effect and CME Maxwell-Chern-Simons Theory B −∂ E j c ˙ E ∇× B − P × ∂ t = ∇⋅ E = c P ⋅ B P = ∂ E ∂ B ∇× ∂ t = 0 ∇⋅ B = 0 Induced Electric Current j = c ˙ E B − P × Induced Electric Charge q = c P ⋅ B =− c g Witten, Wilczek July 24 2010@Yukawa 26
CME from AdS/QCD Models Chiral Magnetic Current Sakai-Sugimoto Model: Rebhan et al, JHEP 0905, 084 (2009) Lifshytz-Lippert, PRD80, 066005 (2009) Sakai-Sugimoto Model & Reissner-Nordstrom BH: Yee, JHEP 0911, 085 (2009) Soft-wall AdS/QCD: Gorsky-Kopnin-Zayakin, 1003.2293 Confusion and (maybe) a Resolution S CS and Bardeen's counter terms change the CME currents? – Axial gauge fields are not dynamical ones so the counter terms should not be applied. Rubakov (2010) July 24 2010@Yukawa 27
Experimental Status July 24 2010@Yukawa 28
Relativistic Heavy-Ion Collisions Nucleus (Au) Collision Energy per nucleon-nucleon collision = 200GeV p 0 =100GeV, M =1GeV → g ~ 100 Same as the kinetic energy by flying mosquitoes M ~3mg, v ~10cm/s July 24 2010@Yukawa 29
Experimental Observation Brookhaven National Laboratory (Gallery) STAR Detector PHENIX Detector ×~ 100M events July 24 2010@Yukawa 30
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