Effective potential and scattering length of shielding polar molecules

We investigate the effective potential and scattering length of ultracold polar molecules under different shielding techniques. First, we derive the effective potential for two polar molecules in the presence of an elliptical polarization field, combined elliptical and linear polarization fields, and combined elliptical polarization and static fields. The effective potential is then expressed as a sum of a zero-range contact interaction and a long-range dipole-dipole interaction under the Born approximation. We find that the first two shielding methods only partially suppress attractive interactions, while the second method allows for the construction of bound states with different polarization shapes. The last shielding method can achieve complete cancellation of residual attractive forces, which is particularly significant for maintaining quantum degeneracy in ultracold dipolar systems. Our results provide a comprehensive understanding of the effective potential and scattering length of shielding polar molecules, which is crucial for studying many-body physics in ultracold dipolar systems.