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非阿贝尔规范场是构成标准模型的基本单元, 非阿贝尔手征动理学理论是描述标准模型在非平衡体系下手征费米子输运的重要理论工具. 在前期工作中, 我们将非阿贝尔手征动理学方程分解为色空间中的色单态和色多重态等不可约表示形式, 这种分解方式可以让手征动理学方程在色空间的规范变换下具有更简单的变换性质. 然而, 这种分解方式在微观描述色自由度的输运方面可能并不直观和方便. 为了描述色自由度具体输运和演化过程, 本文把前期得到的非阿贝尔手征动理学方程在嘉当韦尔基下进行展开. 本文中通过协变梯度展开的方法将非阿贝尔手征动理学方程展开到1阶, 在嘉当韦尔基下将规范场进行展开, 分布函数分解为对角元素部分和非对角元素部分. 结果显示0阶非对角元素分布函数可以诱导出1阶对角元素分布函数贡献, 0阶对角元素分布函数也可以诱导出1阶非对角元素分布函数的贡献. 非对角元素分布函数之间以及非对角元素与对角元素之间一般都是耦合在一起, 但当规范场只存在对角元素时, 非对角元素与对角元素解耦.
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. In our previous work, we decomposed the non-Abelian chiral kinetic equations into color singlet and multiplet in the $SU(N)$ 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.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 $0$ th-order non-diagonal distribution function could induce the$1$ st-order diagonal Wigner function, and the$0$ th-order diagonal distribution function could also induce the$1$ st-order non-diagonal Wigner function.-
Keywords:
- non-Abelian field /
- Wigner function /
- chiral kinetic equation
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[29] Sun Y, Ko C M 2018 Phys. Rev. C 98 014911
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Google Scholar
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Google Scholar
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