Polarization of massive fermions in a vortical fluid

Fermions become polarized in a vortical fluid due to spin-vorticity coupling. Such a polarization can be calculated from the Wigner function in a quantum kinetic approach. By extending previous results for chiral fermions, we derive the Wigner function for massive fermions up to next-to-leading order in spatial gradient expansion. The polarization density of fermions can be calculated from the axial vector component of the Wigner function and is found to be proportional to the local vorticity $\mathbf{\omega }$. The polarizations per particle for fermions and antifermions decrease with the chemical potential and increase with energy (mass). Both quantities approach the asymptotic value $\hslash \mathbf{\omega }/4$ in the large energy (mass) limit. The polarization per particle for fermions is always smaller than that for antifermions, whose ratio of fermions to antifermions also decreases with the chemical potential. The polarization per particle on the Cooper-Frye freeze-out hypersurface can also be formulated and is consistent with the previous result of Becattini et al. [11,27].

Figure 1

The unintegrated polarization per particle defined in Eq. (51) for (a) fermions and (b) antifermions at momentum $\mathbf{p}$ in the unit of the local vorticity $\hslash \mathbf{\omega }$ as functions of $\beta E_p$ and $\beta \mu$.

Figure 2

The ratio $R$ of polarization per particle in Eq. (55) for fermions to antifermions as a function of $\beta E_p$ and $\beta \mu$.

Figure 3

The integrated polarization per particle $\mathbf{\Pi }\left(x\right)/\rho \left(x\right)$ for (a) fermions and (b) antifermions in the unit of the local vorticity $\hslash \mathbf{\omega }$ as functions of $\beta m$ and $\beta \mu$.

Figure 4

The ratio $R$ of the integrated polarization per particle in Eq. (56) for fermions to antifermions. (a) $R$ as a function of $\beta m$ and $\beta \mu$. (b) $R$ as functions of $\beta m$ at three values $\beta \mu =0.5,1,2$ corresponding to short-dashed, long-dashed, and solid lines, respectively.

Figure 5

(a) The integrated polarization per particle $\mathbf{\Pi }\left(x\right)/\rho$ for massless fermions (solid line) and antifermions (long-dashed line) in the unit of $\hslash \mathbf{\omega }$ as functions of $\beta \mu$. (b) The ratio $R$ of the integrated polarization per particle in Eq. (56) for fermions to antifermions as a function of $\beta \mu$.