Shadow gap in the over-doped (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ compound

The electron band around $M$ point in (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ compound -- completely lifted above the Fermi level for $x > 0.7$ and hence has no Fermi Surface (FS) -- can still form an isotropic s-wave gap ($Delta_e$) and it is the main pairing resource generating an s-wave gap ($Delta_h$) with an opposite sign on the hole pocket around $Gamma$ point. The electron band developing the SC order parameter $Delta_e$ but having no FS displays a {it shadow gap} feature which will be easily detected by various experimental probes such as angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscope (STM). Finally, the formation of the nodal gap $Delta_{nodal}$ with $A_{1g}$ symmetry on the other hole pocket with a larger FS is stabilized due to the balance of the interband pairing interactions from the main hole band gap $Delta_h=+Delta$ and the hidden electron band gap $Delta_e = -Delta$.