Collisional corrections to spin polarization from quantum kinetic theory using Chapman-Enskog expansion
Shuo Fang, Shi Pu
Abstract
We have investigated the collisional corrections to the spin polarization pseudovector <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>δ</a:mi> <a:msup> <a:mrow> <a:mi mathvariant="script">P</a:mi> </a:mrow> <a:mrow> <a:mi>μ</a:mi> </a:mrow> </a:msup> </a:mrow> </a:math> using quantum kinetic theory in Chapman-Enskog expansion. We derive the spin Boltzmann equation incorporating the Møller scattering process. We further consider two distinct scenarios using hard thermal loop approximations for simplification. In scenario (I), the vector charge distribution function is treated as off equilibrium under the validity domain of gradient expansion. Remarkably, the polarization induced by thermal vorticity and shear viscous tensors are modified, but <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mrow> <d:mi>δ</d:mi> <d:msup> <d:mrow> <d:mi mathvariant="script">P</d:mi> </d:mrow> <d:mrow> <d:mi>μ</d:mi> </d:mrow> </d:msup> </d:mrow> </d:math> in this scenario does not depend on the coupling constant. In scenario (II), the vector charge distribution function is assumed to be in local thermal equilibrium. Then collisional corrections <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:mi>δ</g:mi> <g:msup> <g:mrow> <g:mi mathvariant="script">P</g:mi> </g:mrow> <g:mrow> <g:mi>μ</g:mi> </g:mrow> </g:msup> </g:mrow> </g:math> in this scenario are at <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi mathvariant="script">O</j:mi> <j:mo stretchy="false">(</j:mo> <j:msup> <j:mi>ℏ</j:mi> <j:mn>2</j:mn> </j:msup> <j:msup> <j:mo>∂</j:mo> <j:mn>2</j:mn> </j:msup> <j:mo stretchy="false">)</j:mo> </j:math> . Additionally, we evaluate the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>δ</o:mi> <o:msup> <o:mi mathvariant="script">P</o:mi> <o:mi>μ</o:mi> </o:msup> </o:math> using the relaxation time approach for comparative analysis. Our results establish the theoretical framework necessary for future numerical investigations on the interaction corrections to spin polarization.