Spin alignment of vector mesons in local equilibrium by Zubarev’s approach

We compute the 00 element of the spin density matrix, denoted as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>ρ</a:mi> <a:mn>00</a:mn> </a:msub> </a:math> and called the spin alignment, up to the second order of the gradient expansion in local equilibrium by Zubarev’s approach. In the first order, we obtain <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>ρ</c:mi> <c:mn>00</c:mn> </c:msub> <c:mo>=</c:mo> <c:mn>1</c:mn> <c:mo>/</c:mo> <c:mn>3</c:mn> </c:math> , meaning that the contributions from thermal vorticity and shear stress tensor are vanishing. The nonvanishing contributions to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>ρ</e:mi> <e:mn>00</e:mn> </e:msub> <e:mo>−</e:mo> <e:mn>1</e:mn> <e:mo>/</e:mo> <e:mn>3</e:mn> </e:math> appear in the second order of gradients in the Belinfante and canonical cases. We also discuss the properties of the spin density matrix under the time reversal transformation. The effective transport coefficient for the spin alignment induced by the thermal shear stress tensor is T odd in the first order, implying that the first order effect is dissipative.