Chiral Kinetic Theory for Quark Matter

高建华

山东大学（威海）

J.H. Gao and Q. Wang, arXiv:1504.07334J.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:

Chiral Symmetry breaking:

Asymptotic freedom:

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

Chiral Magnetic Effect (CME) Chiral Vortical Effect (CVE)

Strong magnetic fields:

Chirality imbalance:

Large OAM:A+A 200GeV

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)

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 or function:

Wigner equations for massless collisionless particle system in constant EM field:

Wigner functions can be expanded as :

Quantum Transport Equations

Vector and axial vector parts: Scalar, pseudoscalar and tensor parts:

Rewrite Wigner equations by left-hand (s= +1) and right-hand (s= -1) parts

Perturbative Expansion Scheme

The equations can be solved in a very consistent iterative scheme !

Iterative Equation:

One more operator One more order

Derivative and weak field expansion:

The Solution up to the First Order

The 0-th order solutions take the local equilibrium form:

The 0-th order equations:

:Local flow 4-velocity

The first order solution can be given by :

:Temperature

Currents and Energy-Momentum Tensor

All the conservation laws and anomaly can be derived naturally:

Integrate over the momentum

Recall:

CME , CVE, CSE and LPE

CME: CVE:

Local polarization effect:

Chiral separation effect:

LPE should be present in both high and low energy heavy-ion collisions!

STAR collaboration PRL 103 (2009) 251601

Azimuthal Charged-Particle Correlations

Chiral Magnetic Effect

+_

PRD22,3080(1980);78:074033(2008) NPA803,227

Chiral vorticity effect:

Chiral Vorticity Effect

PRL106, 062301(2011); PRL109, 232301(2012)

Consider 3-flavor quark matter (u,d,s), Electric current:

Baryon current:

Chiral Magnetic Conductivity

Chiral Magnetic Conductivity:

HTL/HDL results from chiral kinetic theory:

PRD87,034028(2009), NPB337,569(1990), JHEP0510,056(2005), PRD89,096002(2013)

Wigner equations for massless collisionless system in arbitrary EM field:

Wigner Functions in Arbitrary EM Fields

Triangle Operator:

only act on EM fields

Wigner function Decomposition:

Spherical Bessel Functions

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

HTL / HDL result:

Induced currents:

Chiral Magnetic or Parity –odd Conductivity:

Energy Shift and Magnetic Moment

Effective Energy of Chiral Fermion:

Spin Vector:

Energy Shift:

Magnetic moment of massless fermion:

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:

Effective energy: Berry curvature

Energy shift: Magnetic moment

Energy Shift and Magnetic Moment IIEnergy density with :

Phase-space measure with the Berry curvature:

Effective energy: Berry curvature

Energy shift: Magnetic moment

Energy Shift from Spin-Vorticity CouplingParticle 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.

Thanks for your attention！