Quantum Kinetic Theory

Jian-Hua Gao

Shandong University at Weihai

In collaboration with Zuo-Tang Liang, Shi Pu, Qun Wang and Xin-Nian WangPRL 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

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

Introduction

SQGPPre-equilibrium Hadronization 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:

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:

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:

Unified View of Nucleon StructureMathematically, it is similar to the Wigner function of the nucleon

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

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:

Let us find the solutions near the equilibrium, we can generalize the expansionformalism in hydrodynamics to kinetic theory, treat space-time derivativeand 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

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:

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

The 1-st Order Solution

Iterative equations:

The new kinetic coefficients can be fixed uniquely:

Chiral Anomaly

All the conservation laws and chiral anomaly can be derived naturally:

Integrate over the momentum

CME & CVE

Chiral magnetic effect

Chiral vorticity effect

++__

Strong magnetic fields!

Large OAM: (A+A 200GeV)

Charge Separation at RHICCharge Separation at RHIC

STAR collaboration PRL 103 (2009) 251601

1122

2’2’

Azimuthal Charged-Particle Correlations

Approaches to CME/CVEApproaches to CME/CVE

•Gauge/Gravity DualityGauge/Gravity Duality

ErdmengerErdmenger et.al., JHEP 0901,055(2009); et.al., JHEP 0901,055(2009); Banerjee,Banerjee, et.al., JHEP 1101,094(2011); et.al., JHEP 1101,094(2011);

Torabian and Yee,Torabian and Yee, JHEP 0908,020(2009); JHEP 0908,020(2009); Rebhan, Schmitt and Stricher,Rebhan, Schmitt and Stricher, JHEP1001,026(2010); JHEP1001,026(2010);

Kalaydzhyan and Kirsch,Kalaydzhyan and Kirsch, et.al, PRL 106,211601(2011) …… et.al, PRL 106,211601(2011) ……

•Hydrodynamics with Entropy PrincipleHydrodynamics with Entropy Principle

Son and Surowka,Son and Surowka, PRL 103,191601(2009); PRL 103,191601(2009); Kharzeev and Yee,Kharzeev and Yee, PRD 84,045025(2011); PRD 84,045025(2011);

Pu,Gao and Wang,Pu,Gao and Wang, PRD 83,094017(2011)…… PRD 83,094017(2011)……

•Quantum Field Theory Quantum Field Theory Metlitski and Zhitnitsky,Metlitski and Zhitnitsky, PRD 72,045011(2005); PRD 72,045011(2005); Newman and Son,Newman and Son, PRD 73, 045006(2006); PRD 73, 045006(2006);

Lublinsky and Zahed,Lublinsky and Zahed, PLB 684,119(2010); PLB 684,119(2010); Asakawa, Majumder and Muller,Asakawa, Majumder and Muller, PRC81, PRC81,

064912(2010);064912(2010);Landsteiner,Megias and Pena-Benitez,Landsteiner,Megias and Pena-Benitez, PRL 107,021601(2011); PRL 107,021601(2011);

Hou, Liu and Ren,Hou, Liu and Ren, JHEP 1105,046(2011);…… JHEP 1105,046(2011);……

•Quantum Kinetic ApproachQuantum Kinetic Approach

Gao,Liang, Pu, Q.Wang and X.N. Wang, Gao,Liang, Pu, Q.Wang and X.N. Wang, PRL 109,232301(2012)PRL 109,232301(2012)

Son and YamamotoSon and Yamamoto arxiv:1210.8185; arxiv:1210.8185; Stephanov and YinStephanov and Yin PRL 109,(2012)162001 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

Local Polarization Effect

Local polarization effect

Reversal chiral magnetic effect

LPE should be present in both high and low energy heavy-ion collisions witheither low baryonic chemical potential and high temperature or vice versa.

Since for the 3-flavor quark matter,

With Multiple FlavorsConsider 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:

Semi-Classical Kinetic Equation ?

Semi-Classic Kinetic Equation

Quantum transport equations:

Boltzmann EquationWrite Boltzmann equation in space and time components seperately:

where:

These equation can be obtained from Euler-Lagrange equationfor a charged fermion in EM field without considering the chirality:

Where we can treat (x,p) in equal footing

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

Analogy to magnetic field

• Berry connection

• Berry curvature

• Geometric phase

• Chern-Simons number

• Vector potential

• Magnetic field

• Ahanonrov-Bohm phase

• Dirac monopole

• 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)

4D monopole in momentum spaceThe 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)

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!

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.

Thanks for your attention!