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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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[1] Vilenkin A 1980 Phys. Rev. D 22 3080Google Scholar
[2] Kharzeev D E, McLerran L D, Warringa H J 2008 Nucl. Phys. A 803 227Google Scholar
[3] Fukushima K, Kharzeev D E, Warringa H J 2008 Phys. Rev. D 78 074033Google Scholar
[4] Vilenkin A 1978 Phys. Lett. 80B 150
[5] Kharzeev D, Zhitnitsky A 2007 Nucl. Phys. A 797 67Google Scholar
[6] Erdmenger J, Haack M, Kaminski M, Yarom A 2009 JHEP 0901 055
[7] Banerjee N, Bhattacharya J, Bhattacharyya S, Dutta S, Loganayagam R, Surowka P 2011 JHEP 1101 094
[8] Son D T, Zhitnitsky A R 2004 Phys. Rev. D 70 074018Google Scholar
[9] Metlitski M A, Zhitnitsky A R 2005 Phys. Rev. D 72 045011Google Scholar
[10] Gao J H, Liang Z T, Pu S, Wang Q, Wang X N 2012 Phys. Rev. Lett. 109 232301Google Scholar
[11] Stephanov M A, Yin Y 2012 Phys. Rev. Lett. 109 162001Google Scholar
[12] Son D T, Yamamoto N 2013 Phys. Rev. D 87 085016Google Scholar
[13] Chen J W, Pu S, Wang Q, Wang X N 2013 Phys. Rev. Lett. 110 262301Google Scholar
[14] Manuel C, Torres-Rincon J M 2014 Phys. Rev. D 89 096002Google Scholar
[15] Manuel C, Torres-Rincon J M 2014 Phys. Rev. D 90 076007Google Scholar
[16] Chen J Y, Son D T, Stephanov M A, Yee H U, Yin Y 2014 Phys. Rev. Lett. 113 182302Google Scholar
[17] Chen J Y, Son D T, Stephanov M A 2015 Phys. Rev. Lett. 115 021601Google Scholar
[18] Hidaka Y, Pu S, Yang D L 2017 Phys. Rev. D 95 091901Google Scholar
[19] Mueller N, Venugopalan R 2018 Phys. Rev. D 97 051901Google Scholar
[20] Huang A, Shi S, Jiang Y, Liao J, Zhuang P 2018 Phys. Rev. D 98 036010Google Scholar
[21] Gao J H, Liang Z T, Wang Q, Wang X N 2018 Phys. Rev. D 98 036019Google Scholar
[22] Liu Y C, Gao L L, Mameda K, Huang X G 2019 Phys. Rev. D 99 085014Google Scholar
[23] Lin S, Shukla A 2019 JHEP 6 060
[24] Gao L L, Huang X G 2022 Chin. Phys. Lett. 39 021101Google Scholar
[25] Peng H H, Zhang J J, Sheng X L, Wang Q 2021 Chin. Phys. Lett. 38 116701Google Scholar
[26] Sun Y, Ko C M, Li F 2016 Phys. Rev. C 94 045204
[27] Sun Y, Ko C M 2017 Phys. Rev. C 95 034909Google Scholar
[28] Sun Y, Ko C M 2017 Phys. Rev. C 96 024906Google Scholar
[29] Sun Y, Ko C M 2018 Phys. Rev. C 98 014911Google Scholar
[30] Sun Y, Ko C M 2019 Phys. Rev. C 99 011903Google Scholar
[31] Zhou W H, Xu J 2018 Phys. Rev. C 98 044904Google Scholar
[32] Zhou W H, Xu J 2019 Phys. Lett. B 798 134932Google Scholar
[33] Liu S Y F, Sun Y, Ko C M 2020 Phys. Rev. Lett. 125 062301Google Scholar
[34] Stone M, Dwivedi V 2013 Phys. Rev. D 88 045012Google Scholar
[35] Akamatsu Y, Yamamoto N 2014 Phys. Rev. D 90 125031Google Scholar
[36] Hayata T, Hidaka Y 2017 PTEP 2017 073I01
[37] Mueller N, Venugopalan R 2019 Phys. Rev. D 99 056003Google Scholar
[38] Luo X L, Gao J H 2021 JHEP 11 115
[39] Yang D L 2022 JHEP 06 140
[40] Heinz U W 1983 Phys. Rev. Lett. 51 351Google Scholar
[41] Elze H T, Gyulassy M, Vasak D 1986 Phys. Lett. B 177 402Google Scholar
[42] Elze H T, Gyulassy M, Vasak D 1986 Nucl. Phys. B 276 706Google Scholar
[43] Elze H T, Heinz U W 1989 Phys. Rept. 183 81Google Scholar
[44] Ezawa Z F, Iwazaki A 1982 Phys. Rev. D 25 2681
[45] Ezawa Z F, Iwazaki A 1982 Phys. Rev. D 26 631
[46] Gyulassy M, Iwazaki A 1985 Phys. Lett. B 165 157Google Scholar
[47] Huang X G, Mitkin P, Sadofyev A F, Speranza E 2020 JHEP 10 117
[48] Hattori K, Hidaka Y, Yamamoto N, Yang D L 2021 JHEP 2 1
[49] Lin S 2022 Phys. Rev. D 105 076017Google Scholar
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