Transport Properties in Magnetic Field

Transcription

1 University of Illinois at Chicago/ RIKEN-BNL Research Center The Phases of Dense Matter, July 11-Aug 12 INT, July 28, 2016

2 The magnetic field in heavy-ion collisions

3 In heavy-ion collisions, two magnetic fields from the projectiles overlap along the same direction out of reaction plane eb (300 MeV) 2 T 2, but the life time may be short τ 1 fm/c

4 Astrophysical eb of magnetars is about G (1 MeV) 2, but T 2 eb

5 Transport properties in strong magnetic field We will select the two topics: Chiral magnetic wave (Kharzeev-HUY, arxiv: ) Longitudinal electric conductivity in perturbative QCD in leading log (Work in progress with Koichi Hattori and Shiyong Li)

6 Chiral Magnetic Wave

7 Chiral Magnetic Effect (CME) and Chiral Separation Effect (CSE) (Fukushima-Kharzeev-Warringa, Son-Zhitnitsky, Vilenkin) JV = en c 2π 2 µ A B, JA = en c 2π 2 µ V B Note the < AVV > structure

8 CME+CSE+Hydrodynamics = CMW CMW is the inclusive universal language of CME/CSE in hydrodynamics

9 Why do we have waves? JV = Nce B 2π µ 2 A, JA = Nce B 2π 2 µ V

10 Why do we have waves? JV = Nce B 2π µ 2 A, JA = Nce B 2π 2 µ V

11 Why do we have waves? JV = Nce B 2π µ 2 A, JA = Nce B 2π 2 µ V

12 Why do we have waves? JV = Nce B 2π µ 2 A, JA = Nce B 2π 2 µ V

13 CMW: Sound Waves of Chiral Charges Add and subtract CME and CSE to go to Left/Right-handed chiralities JV = en c 2π 2 µ A B, JA = en c 2π 2 µ V B J R/L 1 2 (J V ± J A ) Then, we have a diagonalization of the CME/CSE JR = en c 4π µ 2 RB, JL = en c 4π µ 2 LB, µ L/R 1 α n L/R Hydro charge conservation µ J µ L/R = 0 gives ( t + v χ ) ( n R = 0, t v χ ) n L = 0, v χ = en c B 4π 2 α Two Independent Uni-directional Propagating Waves!

14 Development of a charge dipole from initial axial charge n A > 0, n V = 0 n R = 1 2 n A, n L = 1 2 n A = n R

15 Development of a charge quadrupole from initial electric charge (Burnier-Kharzeev-Liao-HUY, Gorba-Miransky-Shovkovy) n A = 0, n V > 0 n R = 1 2 n V, n L = 1 2 n V = n R

16 Q: Axial charge is not conserved Conservation law is modified to µ J µ A = 1 τ R n A where the relaxation rate is given by 1 τ R α 5 s log(1/α s )T + α s m 2 q/t (the second term is from the quark mass (Kaplan-Reddy)) The CMW dispersion relation becomes ω = i 2τ s ± (Stephanov-HUY-Yin) 1 4τ 2 s + v 2 χk 2 A transition from CMW (k 1/τ R ) to pure diffusion (k 1/τ R ) τ R is numerically larger than 10 fm with α s = 0.2 and m q = 10 MeV It is okay to neglect this in heavy-ion collisions

17 Q: Electromagnetic field is dynamical Longitudinal Mode: The CMW dispersion becomes (Kharzeev-HUY) ω 2 = v 2 χk 2 + m 2 B where v χ = en c 2π 2 α B, m B e2 N c 2π 2 α B CMW becomes massive Numerically, m 1 B 20 fm with eb m2 π and T 200 MeV, the basic reason is the smallness of α EM It is okay to neglect this in heavy-ion collisions

18 Q: Electromagnetic field is dynamical Transverse Mode: Transverse modes become helicity-polarized via the transition from the fermion helicity (axial charge) into the magnetic helicity. For low enough k, the resulting chiral magneto-hydrodynamics features inverse cascades with turbulence (eg. a recent work by Hirono-Kharzeev-Yin and Yamamoto). Due to the smallness of α EM and µ A /T, the space-time scale k α EM µ A is numerically much larger than 10 fm It is okay to neglect this in heavy-ion collisions, unless µ A /T 1/α EM 100

19 Longitudinal Electric Conductivity in Strong Magnetic Field in Perturbative QCD Work in progress with Koichi Hattori and Shiyong Li

20 We will compute the electric conductivity along the direction of magnetic field J = σ zz E in the limit of eb T 2, which means we can focus on the lowest Landau levels (LLL) pertubative QCD in the deconfined phase, and will further assume α s eb T 2 small enough electric field ee m 2 q to avoid quantum anomaly (Schwinger mechanism) dynamical degrees of freedom are thermal quarks and gluons that are nearly massless neutral quark-gluon plasma

21 The final result at leading log (preliminary) is ( ) σ zz = e 2 qf 3 9 eb 1 π 3 T α 2 F s log(t 2 /α s eb) We will show some technical details in the rest of the talk

22 Lowest Landau Level (LLL) thermal quarks-anti quarks are the charge carriers, which are interacting with themselves and thermal gluons These states move only in 1+1 dimensions and their transverse size is l B 1/ eb. The latter will introduce a "form factor" R(q ) e q 4eB along the transverse direction in the interaction vertex. We will also see a "Schwinger phase" in the vertex 2

23 We introduce a small, but finite quark mass T 2 m 2 q ee to avoid Schwinger mechanism in the massless limit Since the 1+1 dim density of states is p z /(2π), the transverse density of states is eb/(2π), and ṗ z = ee ṅ A = (eb/2π)ṗ z /(2π) = e2 4π 2 E B which is precisely the chiral anomaly (Gribov)

24 Semi-classical motion vs. quantum tunneling quantum tunneling amplitude for pair creation e m 2 q ee However, m 2 q will be neglected in the dispersion relation of the hard modes (p T ) which dominate the charge transports.

25 A hybrid kinetic theory description Introduce a distribution function for each of the LLL states: f α,± (p z, z), where α labels the LLL states and (p z, z) is the longitudinal momentum and position, and ± refers to quark-antiquark.

26 Since the applied electric field is homogeneous, the solution will be independent of α and z. Also by charge conjugation and z-reflection, δf (p z ) = δf + (p z ) = +δf + ( p z ). J = 2e ( eb 2π The longitudinal current will be ) + ( ) dp z eb (2π) ˆp z δf + (p z ) = 4e 2π where ˆp z = sgn(p z ) is the longitudinal velocity 0 dp z (2π) δf +(p z )

27 Boltzmann equation for f α,+ (p z, z) t f α,+ (p z, z) + ż z f α,+ (p z, z) + ṗ z pz f α,+ (p z, z) = C[f ] Homogeneous electric field in static limit gives ṗ z = ee, t = z = 0, and to linear order in E, f α,+ (p z, z) = f eq ( p z ) + δf (p z ) and the Boltzmann equation becomes β(ee)ˆp z f eq ( p z )(1 f eq ( p z )) = C[δf ] where f eq (ɛ) 1/(e βɛ + 1) The key is the collision term C[δf (] We will see a leading log of αs 2 log T 2 α seb )

28 Following the previous studies on the case without magnetic fields (Arnold-Moore-Yaffe), we look at the potentially important t-channel scattering diagrams involving LLL quarks-antiquarks.

29 Feynman Rules of LLL States We choose to work in the Landau gauge A = (0, Bx 1, 0). Then (p 2, p z ) are well-defined quantum numbers. p 2 = α is the label for LLL states. Note that there is no concept of p 1 for LLL fermions The wave functions are localized in x 1 as ( φ p2 = e ip 2x 2 H 0 x 1 p ) 2 eb ψ(x) = 1 L2 L z a p2,p z + h.c. Quantization: ( 2 pz eip 2x 2 +ip zz H 0 x 1 p 2 eb p z,p 2 1 ) u 1+1 (p z ) with {a p2,p z, a p 2,p z } = δ p 2,p 2 δ p z,p z

30 Form Factor and Schwinger Phase Working out the interaction vertex H I = g s d 3 x A µ (x) ψ(x)γ µ ψ(x) Only the longitudinal current exists: µ = t or µ = z The vertex has a form factor and a Schwinger phase (Hattori, Kojo, Su) ig s ɛ µ [ū 1+1 (p z)γ µ u 1+1(p z )]R(q )e i q 1 2eB (p 2+p 2 ) where R(q ) = e q2 /(4eB), q = (q 1, p 2 p 2, p z p z )

31 Examples of Collision Terms C q q [f p2 (p z )] = where p = p p p dq 1 (2π) e i q 1 eb (p 2 p (2π) 3 δ (3) (p + p p p ) f (p)f (p )(1 f (p ))(1 f (p )) 2 ) [R(q )] 2 M d 2 p 2 p z (2π) and M is the usual matrix 2 element with 1+1 dim LLL quarks and 3+1 dim gluon exchange 2

32 A leading log appears from the gluon scatterings M = ig 2 s f abc t a 2(p k )(ɛ ɛ ) Q 2 R(q ) Small Q gives a IR logarithmic divergence, which is screened by the 1-loop self energy from LLL quark loop Q 2 Q 2 + m 2 D, md 2 = g2 s π T RN F (eb/2π)e q 2 2eB

33 I = 1 4 Writing δf = βf eq (1 f eq )δχ C qg [δχ(p z )] = dp z (2E p ) 2 where δi δχ(p z ) d 4 Q (2π) 4 (δχ(p z + q z ) δχ(p z )) 2 (2π)δ(q 0 ˆp z q z )f eq (p)(1 f eq (p + q)) d 3 k (2π) 3 (2E k ) 2 M 2 (2π)δ(q 0 ˆk q) βf eq (k)(1 + f eq (k q)) We do small IR expansion for leading IR log δχ(p z + q z ) δχ(p z ) q z (δχ(p z ))

34 The result is C qg [δχ(p z )] = π ( ) T 12 α2 sn c C R T 2 2 log α s eb [f eq ( p z )(1 f eq ( p z ))δχ (p z )] and the Boltzmann equation becomes (ee)ˆp z f eq ( p z )(1 f eq ( p z )) = π ( ) T 12 α2 sn c C R T 3 2 log α s eb [f eq ( p z )(1 f eq ( p z ))δχ (p z )]

35 δχ(p z ) = We have a nice solution 12 ( πn c C R T 2 αs 2 log T 2 α seb )(ee) ( p z + T ˆp z (1 e β pz ) ) and the conductivity is σ zz = e 2 12 ( ) d R N F eb π 3 N c C R T αs 2 log(t 2 /α s eb) ( ) ( ) = e 2 qf 3 9 eb π 3 T α 2 F s log(t 2 /α s eb) for fundamental quarks for N c = 3

36 Further Directions: Dense system/other phases (e.g. Alford-Nishimura-Sedrakian), Viscosities, Smaller magnetic fields

37 Thank you!