Second Order Chiral Transport Coefficients And Chiral Kinetic Equations

Quark-gluon plasmas have been researched by many people in recent years,as a new state of matter dominated by strong interaction.Quark-gluon plasmas are different from protons and neutrons,which quarks and gluons are released from color confinement and can move freely within them,and behave like strongly coupled fluids in experiment.In particle physics,we usually use heavy-ions collision to create quarkgluon plasmas.There are two main experimental facilities,the Relativistic Heavy Ion Collider(RHIC)in the United States and the Large Hadron Collider(LHC)in Europe.In heavy ion collision,at non-centeral collision,the quark-gluon plasma would have a large angular momentum perpendicular to the reaction plane,and the high velocity heavy ions would also generate a strong magnetic field perpendicular to the reaction plane.People can observe some quantum transport phenomena for example chiral magnetic effects and chiral vortical effects,with large angular momentum and magnetic field.Chiral magnetic effects and chiral vortical effects induced by chiral anomalies,and appear as first-order quantum corrections for currents in quark-gluon plasmas.One way to calculate these chiral effects is to calculate the chiral currents of quark-gluon plasmas with quantum corrections.We use the Wigner function method to calculate these chiral effects,because we can derive currents and energy-momentum tensor by Wigner function conveniently.In this paper,I will introduce the definition of Wigner function for spin 1/2 particles and the derivation of Wigner equations.I will introduce how to use the Cilfford algebra to decompose the 4×4 matrix Wigner equations.In order to derive currents and energymomentum tensor,we assume the mass of quark is zero.It will be much easier to calculate massless quark than massess.We use semi-classical expansion for the Wigner equations,and solve the Wigner equations by iterative method in thermal equilibrium.We get the constrain conditions of thermal equilibrium by Wigner equations at zeroth order.At first order,we obtain the terms of the chiral magnetic effects,the chiral vortical effects and other first order chiral effects in currents,and calculate the energymomentum tensor of the first order,which is in agreement with the other’s results.On this basis,we extend the iterative process from the first order to the second order,and obtain the special solution of the second order Wigner function in equilibrium,and then integrate the second order currents and energy momentum tensor.We have obtain second order pure electromagnetic field term,pure vorticity term and electromagnetic field vorticity coupled term in energy momentum tensors.Especially in the second order currents we obtain coupling terms of electromagnetic field and vorticity,and the Hall flow of the form E × B in vector and axial vector currents.When integrating the pure electromagnetic field term of the energy momentum tensor,the ultraviolet divergence appears.The trace anomaly of the energy momentum tensor is obtained by cut down the ultraviolet divergence by the dimensional regularization.We also discuss the general solution of second order Wigner function in equilibrium by Wigner function method.The currents and energy momentum tensor obtained by the Wigner function satisfy the current conservation and energy momentum conservation.We can also systematicly constrain the transport coefficients of currents and energy momentum tensors in terms of current conservation and energy momentum conservation.We use charge,parity and time inversion symmetry to write the general form of currents and energy-momentum in order of h.For the conservation law,we have used the equations of energy momentum conservation,currents conservation,chiral anomaly and trace anomaly.We have derived the constrain equations of transport equations of second order currents and energy momentum tensor in thermal equilibrium,.We need a chiral kinetic equations,if we want to study chiral effects in nonequilibrium.We obtain the second order chiral kinetic equations in seven dimensional phase space under the semi-classical expansion scheme by using Wigner function method.We have analyzed the transformation rules of the second order distribution function in different frames.In thermal equilibrium,our second order chiral kinetic equation can get the same result as previous.