1. Quantum Kinetic Theory Jian-Hua Gao Shandong University at Weihai In collaboration with Zuo-Tang Liang, Shi Pu, Qun Wang and Xin-Nian Wang PRL 109, 232301(2012), PRD 83, 094017(2011) “The First Sino-Americas Workshop and School on the Bound-State Problem in Continuum QCD ” October 22-26, 2013, USTC, Hefei, AnHui, China

2. Outline Introduction Quantum Transport Equation and How to Solve it. CME, CVE and LPE from Quantum Transport Equation Chiral Kinetic Theory and Berry Monopole Summary

3. Introduction SQGP Pre-equilibriumHadronization Freeze-out QGP produced in high energy heavy-ion collisions at RHIC and LHC can be described very well by hydrodynamics: In order to get more fine information, we need to go to microscopic kinetic theory. The classical Boltzmann equation with the external EM fields:

4. Introduction QCD non-trivial vacuum: Instanton & Sphaleron Chirality imbalance: Chiral current: Classical Transport Quantum Transport K.Fukushma, D.E.Kharzeev, H.J.Warringa PRD78:074033,2008 When quantum effects are relevant, classical kinetic theory is not enough! Chiral anomaly:

5. Classical transport theory: Wigner Functions Wigner operator for the spin-1/2 fermion is given by: Gauge link The ensemble average of Wigner operator: Probability density function Quantum transport theory: Wigner functions D.Vasak, M.Gyulassy, H. Elze Annals Phys. 173 (1987) 462-492 The equation satisfied by Wigner operator:

6. Unified View of Nucleon Structure Mathematically, it is similar to the Wigner function of the nucleon

7. Polarized Nucleon & Chiral Fluid Polarized Nucleon: Microscopic chiral system Chiral fluid: Macroscopic chiral system: Hopefully, we expect our quantum transport approach can also give some help for studying the Wigner functions of hadrons. Physically, chiral fluid is not different from the nucleon too far away

8. 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:

9. Let us find the solutions near the equilibrium, we can generalize the expansion formalism in hydrodynamics to kinetic theory, treat space-time derivative and EM field as small magnitudes with the same order. Expand and in powers of and Perturbative Expansion Scheme These equations can be solved in a very consistent iterative scheme ! Iterative equations: 0-th order: 1-st order: One more operator One more order

10. The 0-th Order Solution :Electric Chemical Potential : Chiral Chemical Potential The 0-th order solutions take the local equilibrium form: The 0-th order equations:

11. The 1-st Order Solution Consider the local static solutions The first order solution can be generally made from: Constraint conditions Evolution equations : Local flow 4-velocity

12. The 1-st Order Solution Iterative equations: The new kinetic coefficients can be fixed uniquely:

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

14. CME & CVE Chiral magnetic effect Chiral vorticity effect + _ Strong magnetic fields! Large OAM: (A+A 200GeV)

15. Charge Separation at RHIC STAR collaboration PRL 103 (2009) 251601 1 2 2’ Azimuthal Charged-Particle Correlations

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, JHEP 1105,046(2011);…… Quantum Kinetic ApproachQuantum Kinetic Approach Gao,Liang, Pu, Q.Wang and X.N. Wang, PRL 109,232301(2012) Gao,Liang, Pu, Q.Wang and X.N. Wang, PRL 109,232301(2012) Son and Yamamoto arxiv:1210.8185; Stephanov and Yin PRL 109,(2012)162001 Son and Yamamoto arxiv:1210.8185; Stephanov and Yin PRL 109,(2012)162001 The first time to get both CME and CVE in kinetic theory. CME was first introduced by K.Fukushma, D.E.Kharzeev, H.J.Warringa PRD78:074033,2008

17. Local Polarization Effect 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.

18. Since for the 3-flavor quark matter, With Multiple Flavors Consider 3-flavor quark matter (u,d,s), Vector current: D.Kharzeev and D.T.Son, PRL106, 062301(2011); J.H.Gao, Z.T.Liang, S.Pu, Q.Wang, X.N. Wang, PRL109, 232301(2012) Axial current: Baryonic: Electric:

19. Semi-Classical Kinetic Equation ? Semi-Classic Kinetic Equation Quantum transport equations:

20. Boltzmann Equation Write Boltzmann equation in space and time components seperately: where: These equation can be obtained from Euler-Lagrange equation for a charged fermion in EM field without considering the chirality: Where we can treat (x,p) in equal footing

21. Phase space description of charged fermion The action of the chiral fermion ( for exmaple, helicity +1 particle) M.A. Stephanov, Y. Yin, PRL 109 (2012) 162001 Berry curvature: Berry connection: EOM can be derived from Euler-Lagrange equation Berry Monopole: Berry monopole is responsible for chiral anomaly, CME and CVE

22. Analogy to magnetic field Berry connection Berry curvature Geometric phase Chern-Simons number Vector potential Magnetic field Ahanonrov-Bohm phase Dirac monopole

23. It is the first time to obtain covariant chiral kinetic equation in 4D The result is determined by the singular 4-vector: Covariant Chiral Kinetic Equation in 4D can be obtained by rearranging the equations for vector and axial vector components of Wigner functions: where J.W. Chen, S.Pu, Q.Wang, X.N. Wang, PRL 110, 262301(2013) Covariant Chiral Kinetic Equation in 4D (CCKE)

24. 4D monopole in momentum space The singular 4-vector together with the on-shell leads to chiral anomaly, which can be shown by taking divergence of the right-handed or left-handed current: 4D Berry monopole in Euclidean space : J.W. Chen, S.Pu, Q.Wang, X.N. Wang, PRL 110, 262301(2013)

25. The chiral kinetic equation in 3-dimensions by integration over for the covariant chiral kinetic equation as Derivation of 3D Chiral Kinetic Equation D.T. Son, N. Yamamoto, PRL 109 (2012) 181602 M.A. Stephanov, Y. Yin, PRL 109 (2012) 162001 Berry monopole from 4D to 3D J.W. Chen, S.Pu, Q.Wang, X.N. Wang, PRL 110, 262301(2013) Vorticity terms come naturally from the covariant chiral kinetic eqution!

26. Summary A consistent iterative scheme to solve quantum transport equations has been set up. Chiral anomaly, CME and CVE are natural results of quantum transport theory. A local polarization effect due to the vorticity can be expected in non-central heavy ion collisions. Berry monopole and covariant chiral kinetic equation can be obtained directly from Wigner equation.

27. Thanks for your attention!