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Quantum Kinetic Theory

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).

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Quantum Kinetic Theory

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

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

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

  7. Polarized Nucleon & Chiral Fluid Physically, chiral fluid is not different from the nucleon too far away 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.

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

  9. Perturbative Expansion Scheme 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 0-th order: 1-st order: One more order One more operator Iterative equations: These equations can be solved in a very consistent iterative scheme !

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

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

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

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

  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 2 1 2’ Azimuthal Charged-Particle Correlations

  16. Approaches to CME/CVE CME was first introduced by K.Fukushma, D.E.Kharzeev, H.J.Warringa PRD78:074033,2008 • Gauge/Gravity Duality • 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); • Kalaydzhyan and Kirsch, et.al, PRL 106,211601(2011) …… • Hydrodynamics with Entropy Principle • Son and Surowka, PRL 103,191601(2009); Kharzeev and Yee, PRD 84,045025(2011); • Pu,Gao and Wang, PRD 83,094017(2011)…… • Quantum Field Theory • 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, • 064912(2010);Landsteiner,Megias and Pena-Benitez, PRL 107,021601(2011); • Hou, Liu and Ren, JHEP 1105,046(2011);…… • Quantum Kinetic Approach • 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 The first time to get both CME and CVE in kinetic theory.

  17. Local Polarization Effect Reversal chiral magnetic effect Local polarization 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. With Multiple Flavors Consider 3-flavor quark matter (u,d,s), Vector current: Baryonic: Electric: Axial current: Since for the 3-flavor quark matter, 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)

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

  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 M.A. Stephanov, Y. Yin, PRL 109 (2012) 162001 • The action of the chiral fermion ( for exmaple, helicity +1 particle) Berry connection: • EOM can be derived from Euler-Lagrange equation • Berry curvature: • 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. Covariant Chiral Kinetic Equation in 4D (CCKE) • Covariant Chiral Kinetic Equation in 4D can be obtained by rearranging the equations for vector and axial vector components of Wigner functions: • where • It is the first time to obtain covariant chiral kinetic equation in 4D • The result is determined by the singular 4-vector: J.W. Chen, S.Pu, Q.Wang, X.N. Wang, PRL 110, 262301(2013)

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

  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!

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