Photons in the Chiral Magnetic Effect

Transcription

1 Photons in the Chiral Magnetic Effect Kenji Fukushima Department of Physics, Keio University June 25, CPODD 1

2 Current from the Quantum Anomaly Anomaly Relation j = N c i=flavor Q i 2 e 2 μ 5 2π 2 B Chiral Magnetic Effect = QCD anomaly QED anomaly No correction from higher-order fluctuations (apart from the RG non-invariance) Vilenkin (1980) Rediscovered over and over again! June 25, CPODD 2

3 Conventional Explanation Classical Picture Left-handed Quarks = momentum anti-parallel to spin B Right-handed Quarks = momentum parallel to spin Kharzeev-McLerran-Warringa (2007) Fukushima-Kharzeev-Warringa (2008) J 0 if N 5 = N R N L 0 June 25, CPODD 3

4 Conventional Interpretation Seemingly, massless and deconfined quarks flow in response to the external magnetic field. Signature for chiral restoration and deconfinement Chiral restoration is not a necessary condition. CME formula is insensitive to IR quantities BUT chiral restoration is necessary for a large N 5 Deconfinement is not a necessary condition. Charged hadrons can generate an electric current BUT more topological excitations at high T June 25, CPODD 4

5 Questions What is the CME in the hadronic phase? What is the origin and the meaning of µ 5? What is really necessary for the CME? What is the crucial difference driven by space-time inhomogeneous backgrounds? June 25, CPODD 5

6 What is the CME in the hadronic phase? June 25, CPODD 6

7 Calculation in the Hadronic Phase Chiral Effective Lagrangian L χ = f 2 π 4 tr [ μ U μ U +2 B( MU +UM )] chiral condensate NG boson mass term quark mass If U =e i πa t a / f π is plugged in L χ = 1 2 [ μ π a μ π a m π 2 π a π a + ] June 25, CPODD 7

8 Not Sufficient Anomaly Term θ dependence L U (1)A = N f χ top [ 2 θ i ]2 2 tr (ln U ln U ) Di Vecchia-Veneziano (1980) Correctly capture the coupling between θ and U U(1) A symmetry is explicitly broken η' becomes massive (Witten-Veneziano mass formula at large N c ) June 25, CPODD 8

9 U(1) A Rotation Chiral Rotation q R V R q R q L V L q L U V R U V L U(1) A Rotation V R exp(i α) V L exp( i α) θ θ 2 N f α θ i 2 tr (lnu ln U ) is invariant June 25, CPODD 9

10 Not Yet Sufficient Local Chiral Rotations Theory can be invariant with external vector fields: D μ U = μ i [v μ, U ] i {a μ,u }+ i 2 ( μθ+2 tr(a μ ))U v μ +a μ V (v R μ +a μ +i μ )V R v μ a μ V (v L μ a μ +i μ )V L Local U(1) A Rotation QCD with v µ, a µ Effective theory with v µ, a µ Same anomaly Bardeen (1969) June 25, CPODD 10

11 Wess-Zumino-Witten Action Simple Form in Two-Flavor (Kaiser 2001) L WZW = N c 32 π 2 ϵμ νρ σ [ tr {U r μ U l ν r μ l ν π 0 2 γ +i Σ μ (U r ν U + l ν )} tr (v ρσ )+ 2 3 tr(σ μ Σ ν Σ ρ ) tr(v σ )] r μ =v μ +a μ l μ =v μ a μ Σ μ ν =U μ U v μ ν = μ v ν ν v μ i[v μ, v ν ] r μ =r μ 1 2 tr (r μ) lμ =l μ 1 2 tr (l μ) June 25, CPODD 11

12 Electromagnetic Vector Potential Choice of the Vector Fields v μ =eq A μ a μ =0 Q : Electric-charge Matrix L WZW = N c tr (Q) 32 π 2 ϵ μ νρσ { i e 2 tr [(Σ μ + Σ μ ) τ 3 ] A ν ρ A σ π 0 2 γ 2e 3 tr(σ μ Σ ν Σ ρ ) A σ } June 25, CPODD 12

13 Electric Current I j μ ( x)= δ d 4 x L δ A μ ( x) L χ j μ χ = i e f 2 π 4 tr[(σμ Σ μ )τ 3 ] e 2 (π i μ π + π + i μ π )+ Non-anomalous part just gives the pion current June 25, CPODD 13

14 Electric Current II μ L WZW j WZW = N c tr(q) ϵ μ νρ σ (~ ~ ~ ~) 32 π 2 N c tr(q)e 2 8π 2 f π ϵ μ νρσ ( ν π 0 ) F ρ σ + π 0 -domain wall (Son-Stephanov 2007) Skyrmion (Hashimoto et al 2011) Similar to the CME but not exactly it June 25, CPODD 14

15 Still Not Sufficient Contact Contribution (RG non-invariant) N c { L P = 8 N f π 2 ϵμ νρσ tr [ v μ( νv ρ 2i 3 v νv ρ )] σ θ +tr(a μ D ν v a ρ )( 4 3 tr(a σ)+ σ θ) 2 3 N f tr(a μ ) tr( ν a ρ ) σ θ} = N ce 2 tr(q 2 ) 8 N f π 2 ϵ μ νρσ A μ ( ν A ρ ) σ θ c.f. Gorbar-Miransky-Shovkovy Fukushima-Ruggieri L P j P μ = N c e 2 tr (Q 2 ) 4 N f π 2 ϵ μ νρσ ( ν A ρ ) σ θ CME June 25, CPODD 15

16 What is the origin and the meaning of µ 5? June 25, CPODD 16

17 = Space-time Dependent θ j P μ = N ce 2 tr(q 2 ) 4 N f π 2 ϵ μ νρ σ ( ν A ρ ) σ θ μ 5 = 0θ(t) 2 N f j P = N c e 2 tr(q 2 ) 2 π 2 μ 5 B June 25, CPODD 17

18 Origin of Space-time Dependent θ L χ = f 2 π 4 tr [ μ U μ U +2 B( MU +UM )] L U (1)A = N f χ top 2 [ θ i 2 tr (ln U ln U ) Vacuum state should minimize the energy U e i η/ f η θ+η/ f η =θ eff =0 June 25, CPODD 18 ]2 If BM = 0 (small enough) θ-dep is gone When η condenses, it is absorbed in the redefinition of (Normalization condition of U) c.f. linear-σ model (Witten 1980)

19 Chiral Symmetry Breaking Chiral Condensate and Chiral Circle σ ψψ π ψi γ 5 τ ψ June 25, CPODD 19

20 Chiral Symmetry Breaking Chiral Condensate and Chiral Circle σ ψψ π ψi γ 5 τ ψ Disoriented Chiral Condensate (DCC) June 25, CPODD 20

21 Chiral Symmetry Breaking Chiral Condensate and Chiral Circle σ ψψ η 0 ψi γ 5 ψ Local Parity Violation (LPV) June 25, CPODD 21

22 Local Parity Violation (LPV) Heavy-Ion Hot and Dense Matter Heavy-Ion + η 0 >0 η 0 <0 η 0 <0 η 0 >0 η 0 >0 η 0 <0 + CME is certainly a signature for the LPV. LPV or η-dcc needs the chiral symmetry restoration. Thus, CME could be an indirect evidence for it. Confinement-deconfinement is completely irrelevant. June 25, CPODD 22

23 What is the crucial difference driven by inhomogeneous backgrounds? June 25, CPODD 23

24 Contact Term Again L P = N c e 2 tr(q 2 ) 8 N f π 2 ϵ μ νρ σ A μ ( ν A ρ ) σ θ What is the natural interpretation of this? Three-point (θ-γ-γ) vertex Split the vector potential to the external B and the photon A=Ā+δ A June 25, CPODD 24

25 Photon Vertices L P = N ce 2 tr(q 2 ) 8 N f π 2 ϵ μ νρ σ [δ A μ ( ν δ A ρ )+δ A μ F νρ ] σ θ θ-γ-γ θ-b-γ Because eb is as large as the QCD scale, only the second term is the dominant process. Chiral Magnetic Effect = Primakoff Effect γ+γ * (neutral field) June 25, CPODD 25

26 Reverse Primakoff Effect Neutral field provided by the LPV (η condensate) q 0 dn γ d 3 q = q 2 α eζ(q) 2(2π) 3 q π 3 ζ(q)= d 4 x e i q x eb ( x)μ 5 ( x) 2 Momentum Conservation Calculable Unknown c.f. Basar-Kharzeev-Skokov (Trace Anomaly) June 25, CPODD 26

27 Characteristics Small, but not hopelessly small... q 0 (d N γ /d 3 q) GeV 2 Emitted to the directions perpendicular to B. Competing with other (pure-b) effects Hiroshima Group Photon's energy and momentum reflect the typical distribution of the LPV domains. Homogeneous backgrounds cannot supply the necessary energy and momentum. June 25, CPODD 27

28 Summary June 25, CPODD 28

29 Summary What is the CME? What flows to generate the current with hadrons. Microscopic origin of the LPV needs much more investigations (DCC-type sumulation in the hadron phase and/or Glasma simulation with isotropization.) Photon related to the strong-b background need more investigations. Anomalous and non-anomalous processes both are important. June 25, CPODD 29