1. Chiral Anomaly and Local Polarization Effect from Quantum Transport Approach 高建华 山东大学（威海） J.H. Gao, Z.T. Liang, S. Pu, Q. Wang, X.N. Wang, PRL 109, 232301(2012)

2. Outline Introduction Chiral Anomalous Fluid Chiral Anomalous Fluid from Quantum Transport Approach Summary

3. Quantum Chromo Dynamics QCD : Quark Confinement: Chiral Symmetry breaking: Asymptotic freedom:

4. Instanton and Sphaleron Gluon field configuration with topological winding number: Lepton and baryon number non-conservation Strong CP violation

5. Anomalous fluid in HIC Hydrodynamics with EM fields: SphaleronChirality imbalance:Chiral current: Hydrodynamic equations with anomalous axial current :

6. Hydrodynamics with Chiral Anomalies Hydrodynamic Equation with Anomalies : D.T. Son, P. Surowka PRL103:191601,2009 Requiring, The new kinetic coefficients can be fixed uniquely:

7. From one current to two currents: Requiring S. Pu, J.H. Gao, Q. W. PRD83:094017,2011 Hydrodynamics with Chiral Anomalies

8. CME & CVE Chiral magnetic effectChiral vortical effect + _ STAR collaboration PRL 103 (2009) 251601 K.Fukushma, D.E.Kharzeev, H.J.Warringa PRD78:074033,2008 (A+A 200GeV)

9. Classical transport approach Gauge invariant Wigner operator: Gauge link The ensemble average of Wigner operator: Probability density function Quantum transport approachWigner functions D.Vasak, M.Gyulassy, H. Elze Annals Phys. 173 (1987) 462-492 From Macroscopic to Microscopic

10. Wigner equations for massless collisionless particle system in homogeneous background EM field : Wigner functions can be expanded as : Quantum Transport Equations Vector parts:Scalar and tensor parts:

11. In order to find the solutions near the equilibrium, we can expand and in powers of and Perturbative Expansion Scheme The equations can be solved in a very consistent iterative scheme ! Iterative equations: 0-th order: 1-st order: One more operatorOne more order

12. The 0-th Order Solution The 0-th order solutions take the local equilibrium form: The 0-th order equations: :Local flow 4-velocity

13. The 1-st Order Solution Consider the local static solutions The first order solution can be generally made from Constraint conditions for Vlasov equation Evolution equations for ideal fluids Iterative equations:

14. Chiral Anomaly All the conservation laws and anomaly can be derived naturally: Integrate over the momentum

15. CME, CVE, LPE CME:CVE: Local polarization effect Reversal chiral magnetic effect LPE should be present in both high and low energy heavy-ion collisions with either low baryonic chemical potential and high temperature or vice versa.

16. Approaches to CME/CVE Gauge/Gravity DualityGauge/Gravity Duality Erdmenger et.al., JHEP 0901,055(2009); Banerjee, et.al., JHEP 1101,094(2011); Erdmenger et.al., JHEP 0901,055(2009); Banerjee, et.al., JHEP 1101,094(2011); Torabian and Yee, JHEP 0908,020(2009); Rebhan, Schmitt and Stricher, JHEP1001,026(2010); Torabian and Yee, JHEP 0908,020(2009); Rebhan, Schmitt and Stricher, JHEP1001,026(2010); Kalaydzhyan and Kirsch, et.al, PRL 106,211601(2011) …… Kalaydzhyan and Kirsch, et.al, PRL 106,211601(2011) …… Hydrodynamics with Entropy PrincipleHydrodynamics with Entropy Principle Son and Surowka, PRL 103,191601(2009); Kharzeev and Yee, PRD 84,045025(2011); Son and Surowka, PRL 103,191601(2009); Kharzeev and Yee, PRD 84,045025(2011); Pu,Gao and Wang, PRD 83,094017(2011)…… Pu,Gao and Wang, PRD 83,094017(2011)…… Quantum Field TheoryQuantum Field Theory Metlitski and Zhitnitsky, PRD 72,045011(2005); Newman and Son, PRD 73, 045006(2006); Metlitski and Zhitnitsky, PRD 72,045011(2005); Newman and Son, PRD 73, 045006(2006); Lublinsky and Zahed, PLB 684,119(2010); Asakawa, Majumder and Muller, PRC81, Lublinsky and Zahed, PLB 684,119(2010); Asakawa, Majumder and Muller, PRC81, 064912(2010);Landsteiner,Megias and Pena-Benitez, PRL 107,021601(2011); 064912(2010);Landsteiner,Megias and Pena-Benitez, PRL 107,021601(2011); Hou, Liu and Ren, JHEP 1105,046(2011); Hou, Liu and Ren PRD86(2012)121703…… Hou, Liu and Ren, JHEP 1105,046(2011); Hou, Liu and Ren PRD86(2012)121703…… Quantum Kinetic ApproachQuantum Kinetic Approach Stephanov and Yin PRL 109,(2012)162001, Son and Yamamoto PRD 87 (2013) 8, 085016; Stephanov and Yin PRL 109,(2012)162001, Son and Yamamoto PRD 87 (2013) 8, 085016; Chen, Pu, Q.Wang and X.N. Wang, PRL 110 (2013)262301 Chen, Pu, Q.Wang and X.N. Wang, PRL 110 (2013)262301

17. Summary A consistent iterative scheme to solve quantum transport equations has been set up, in which transport approach, hydro expansion and chiral anomaly are totally consistent with each other. Induced currents by vorticity and magnetic field are natural results of quantum transport theory. A local polarization effect due to the vorticity can be expected in non-central heavy ion collisions. The success of this approach urges us to go further: Non-interacting systemInteracting system, QGP First orderSecond order, … Constant EM fieldArbitrary EM field

18. 谢谢！