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

Luo Xiao-Li; Gao Jian-Hua

doi:10.7498/aps.72.20222471

Abstract

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:

Authors and contacts

Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Shandong University, Weihai 264209, China

Corresponding author: Gao Jian-Hua, gaojh@sdu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11890710, 11890713, 12175123)

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References

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Cited By

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

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