Chiral Kinetic Theory for Quark Matter 高建华 山东大学（威海） J.H. Gao and Q. Wang, arXiv:1504.07334 J.H. Gao, Z.T. Liang, S. Pu, Q. Wang and X.N. Wang, PRL 109, 232301(2012) “2015手征有效场论研讨会”，2015年8月14日-18日，山东威海

Outline Introduction Vector and Axial Currents Induced by Magnetic fields and Vorticity Magnetic Moment and Spin-Vorticity Coupling of Chiral Fermions Summary

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

Gluon field configuration with topological winding number: Instanton & Sphaleron Gluon field configuration with topological winding number: PRD 28,2019; PRD 30,2212; PLB 155,36; PRD 36,581; NPB 308,885; PRD 37,1020; PRD 43,2027; PLB 326,118 …

CME & CVE Chirality imbalance: Strong magnetic fields: Large OAM: A+A 200GeV Large OAM: Chiral Magnetic Effect (CME) Chiral Vortical Effect (CVE) PRD22,3080(1980);78:074033(2008);NPA803,227 PRL106, 062301(2011); PRL109, 232301(2012)

Approaches to CME/CVE 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); Hou, Liu and Ren PRD86(2012)121703…… Quantum Kinetic Approach 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, J.Y. Chen, Son and Stephanov PRL115, 021601 (2015)

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 or function: D.Vasak, M.Gyulassy, H. Elze Annals Phys. 173 (1987) 462-492

Quantum Transport Equations Wigner equations for massless collisionless particle system in constant EM field: Wigner functions can be expanded as : Vector and axial vector parts: Scalar, pseudoscalar and tensor parts:

Perturbative Expansion Scheme Rewrite Wigner equations by left-hand (s= +1) and right-hand (s= -1) parts Derivative and weak field expansion: One more operator One more order Iterative Equation: The equations can be solved in a very consistent iterative scheme !

The Solution up to the First Order The 0-th order equations: The 0-th order solutions take the local equilibrium form: The first order solution can be given by : :Local flow 4-velocity :Temperature

Currents and Energy-Momentum Tensor Recall: Integrate over the momentum All the conservation laws and anomaly can be derived naturally:

CME , CVE, CSE and LPE CME: CVE: Chiral separation effect: Local polarization effect: LPE should be present in both high and low energy heavy-ion collisions!

Chiral Magnetic Effect STAR collaboration PRL 103 (2009) 251601 + _ PRD22,3080(1980);78:074033(2008) NPA803,227 Azimuthal Charged-Particle Correlations

Chiral Vorticity Effect PRL106, 062301(2011); PRL109, 232301(2012) Consider 3-flavor quark matter (u,d,s), Electric current: Baryon current:

Chiral Magnetic Conductivity HTL/HDL results from chiral kinetic theory: PRD87,034028(2009), NPB337,569(1990), JHEP0510,056(2005), PRD89,096002(2013)

Wigner Functions in Arbitrary EM Fields Wigner equations for massless collisionless system in arbitrary EM field: Spherical Bessel Functions Triangle Operator: only act on EM fields Wigner function Decomposition: Left-hand (s= +1) and right-hand (s= -1) parts

Linear Response Theory for Wigner Functions Weak Field Approximation: Zero-th Order Equation:

Linear Response Theory for Wigner Functions The first Order Equation: Formal solution: Parity –odd part of the Wigner function in momentum space:

Chiral Magnetic Conductivity Induced currents: Chiral Magnetic or Parity –odd Conductivity: HTL / HDL result:

Energy Shift and Magnetic Moment Effective Energy of Chiral Fermion: Energy Shift: Magnetic moment of massless fermion: Spin Vector: Son and Yamamoto, PRD87,085016(2013) Gao and Wang 1504.07334

Energy Shift and Magnetic Moment I Particle density with : Phase-space measure with the Berry curvature: Berry curvature Effective energy: Energy shift: Magnetic moment

Energy Shift and Magnetic Moment II Energy density with : Phase-space measure with the Berry curvature: Effective energy: Berry curvature Energy shift: Magnetic moment

Energy Shift from Spin-Vorticity Coupling Particle density and energy density with : Effective energy: Energy shift: Spin vector

Summary A consistent iterative scheme to solve Wigner equations has been set up. Wigner functions can describe CME, CVE, LPE, magnetic energy shift and spin-vorticity coupling in a very consistent way, Up to now, it is the only chiral kinetic approach that could give the result of one-loop parity-odd conductivity. All these successes demonstrate that Wigner functions capture comprehensive aspects of physics for chiral fermions in EM fields. More interesting results are expected from Wigner functions.

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