Non-Abelian chiral kinetic equations in the Cartan-Weyl basis

Abstract

<sec>Non-Abelian gauge field is the fundamental element of the standard model. Non-Abelian chiral kinetic theory can be used to describe how the chiral fermions in standard model transport in a non-equilibrium system. </sec><sec>In our previous work, we decomposed the non-Abelian chiral kinetic equations into color singlet and multiplet in the <inline-formula><tex-math id="M1">\begin{document}$SU(N)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.png"/></alternatives></inline-formula> color space. In this formalism, the chiral kinetic equations preserve the gauge symmetry in a very apparent way. However, sometimes we need to describe the microscopic process of the specific color degree, e.g. the color connection in the hadronization stage. In order to describe such a process, it will be more convenient to decompose the non-Abelian chiral kinetic equations in the Cartan-Weyl basis. </sec><sec>In this work, we choose the matrix elements of the Wigner function in fundamental representation of color space as the direct variables and decompose the gauge field or strength tensor field in the Cartan-Weyl basis. By using the covariant gradient expansion, we decompose the non-Abelian chiral kinetic equations into the coupled kinetic equations for diagonal distribution function and non-diagonal distribution function up to the first order. When only diagonal elements exist in the gauge field with non-diagonal elements and diagonal elements decoupled, the non-Ableian chiral kinetic equation will be reduced to the form in the Abelian case. When the non-diagonal elements of the gauge field are present, the kinetic equations are totally tangled between diagonal distribution function and non-diagonal distribution function. Especially, the <inline-formula><tex-math id="M2">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.png"/></alternatives></inline-formula>th-order non-diagonal distribution function could induce the <inline-formula><tex-math id="M3">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.png"/></alternatives></inline-formula>st-order diagonal Wigner function, and the <inline-formula><tex-math id="M4">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.png"/></alternatives></inline-formula>th-order diagonal distribution function could also induce the <inline-formula><tex-math id="M5">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.png"/></alternatives></inline-formula>st-order non-diagonal Wigner function. </sec>