  We theoretically explore the superconducting critical temperature of hole
doped blue phosphorene. Implementing the density functional theory
calculations, we show that for the hole doped blue phosphorene, the isotropic
superconducting state is induced owing to the quite strong electron-phonon
coupling. The theory is based on the Migdal-Eliashberg formalism and the
critical temperature is obtained through set-of-equations, self-consistency. In
addition, we include a vertex correction diagram to the Migdal-Eliashberg
formalism. The inclusion of the vertex correction beyond the Migdal-Eliashberg
formalism changes the $T_c$ about $\pm20$K, depending on the level of the
doping. Our accurate numerical results show that the superconducting critical
temperature is still quite high, even in the cases that the vertex correction
is implemented.
