The electron electric-dipole-moment (eEDM) is a powerful tool for exploring new particles. The candidates for eEDM search are heavy atoms and their molecules, which are well known for the obvious relativistic effect. Lead atom is considered to be the most ideal relativistic atom [Park et al., Nat. Commun. 11(1), 815 (2020)]. PbH molecule is an important representative of the Pb compound and is considered a cold candidate molecule due to the high diagonal Franck-Condon factors. We systematically investigated the (eEDM) searches of PbH using a two-component approach. The parity- and time-reversal symmetry violation constants of ground and excited states, including internal effective electric field Eeff, electron-nucleon scalar-pseudoscalar interaction constant WP,T, and nuclear magnetic quadrupole moment, were obtained and compared to other molecules. In addition, we designed two experimental methods to measure the sensitivity of the eEDM, indicating that the Faraday rotation method could greatly improve its sensitivity.
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