Interstellar Magnetic Field Effects on the Asymmetric Structure of Pulsar Halos

There has been ongoing debate about the potential unconfirmed asymmetric structure of the diffuse $\gamma$-ray emission of the Geminga halo. In this work, we adhere to first principles, injecting and propagating individual cosmic ray (CR) electrons in 3D realizations of turbulent magnetic fields characterized by Kolmogorov turbulence and Bohm turbulence. The particle motion is governed by the Lorentz force, and their energy losses are accounted for through synchrotron and inverse Compton scattering. Furthermore, we consider potential regular magnetic field and the inclination angle between the line of sight (LOS) and the magnetic field lines (MFLs) direction, calculating the resulting gamma-ray emission, comparing it with the HAWC surface brightness measurements. We confirmed that the coherence length $L_{\rm c}$ could be constrained around 1 pc as previous work suggested, the $\chi^2$/d.o.f. fitting findings using the HAWC data suggest that the presence of a regular magnetic field has a marginal improvement over a purely turbulent magnetic field. The Bohm turbulence corresponds to the cosmic-ray-driven instability scenario, which is characterized by more tangled magnetic field lines and a more isotropic distribution of cosmic rays around the injection site, performs worse than Kolmogorov turbulence in reproducing the observed morphology. This results suggest the possibility of potential filamentary structure inside the Geminga halo, while limited by the resolution of very-high-energy (VHE) detectors, the inner asymmetry structure may be invisible. In extreme situation, such as a large $L_{\rm c}$ is superimposed with a strong regular magnetic field, when the LOS is coincidently parallel to the direction of the MFL, the resulting pulsar halo morphology is overall isotropic. However, our results suggest that even in this scenario, the intrinsic filamentary structure would not be disrupted.