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

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

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

4. 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 …

5. 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, (2011); PRL109, (2012)

6. 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, (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, ;
Chen, Pu, Q. Wang and X.N. Wang, PRL 110 (2013)262301,
J.Y. Chen, Son and Stephanov PRL115, (2015)

7. 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)

8. 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:

9. 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 !

10. 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

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

12. 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!

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

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

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

16. 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

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

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

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

20. 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

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

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

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

24. 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.

25. Thanks for your attention！