Synchrotron polarization of anisotropic electron distribution in GRB prompt emission

In gamma-ray bursts (GRBs), the electron pitch angle ($\alpha$) is usually assumed to be isotropically distributed. However, recent numerical simulations indicate that only the high-energy electrons (with Lorentz factors $\gamma>\gamma_{iso}$) are distributed isotropically, whereas the low-energy electrons (with $\gamma<\gamma_{iso}$) follow an energy-dependent anisotropic distribution during magnetic reconnection. The mean value of $\sin^2 \alpha$ approximately follows the relation $\langle \sin^2 \alpha \rangle \propto \gamma^{m}$ for $\gamma<\gamma_{iso}$. In principle, polarization measurements may help us constrain the pitch-angle distribution of electrons in GRBs, since different pitch-angle distributions produce distinct synchrotron polarization signatures. The polarization of GRBs produced by isotropically distributed electrons has been extensively studied. In this paper, we investigate synchrotron polarization produced by anisotropically distributed electrons within a globally toroidal magnetic field in GRB prompt emission. Our results show that the synchrotron PDs in the $\gamma$-ray and X-ray bands produced by anisotropically distributed electrons are systematically lower than those produced by isotropically distributed electrons, while the PD in the optical band could be either lower or higher than that of isotropically distributed electrons, depending primarily on the value of the energy slope $m$. In addition, we compared our numerical results with observational data, and the comparison suggests that an anisotropic distribution of electrons may offer a potential explanation for the PD and spectral data of some GRBs.