is the surface of the $FeSe$ unit cell and $a$ is the Fe-Fe distance, and the bare pairing $EPI$ potential is $V_{epi}^{0}=2g_{0}^{2}/\Omega$. The coupling constant $\lambda_{m}$ corresponds to $\lambda_{m}$ used in [@Johnston1-2], where the self-energy effects are studied at $T>0$. It is important to point out that $\lambda_{m}$ is (oxygen) *mass-dependent*, contrary to [@Johnston1-2]. Since $\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}$ is *mass-independent* then $\lambda_{m}\sim\Omega^{-1}\sim M^{1/2}$. In the following we discuss the case when $\mathbf{k}=\mathbf{k}_{F}$, i.e. $\xi(\mathbf{k})=0$. For $\omega\ll\Omega$ one has $\Sigma_{epi}(\mathbf{k},\omega)=-\lambda_{m}\omega$ which means that the slope of $\Sigma_{epi}(\mathbf{k},\omega)$ is *mass-dependent*. The latter property can be measured by $ARPES$ and thus the $EPI-FSP$ theory can be tested. Note, that in the $EPI-FSP$ theory the critical temperature $T_{c0}(=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/4)$ - see details below, is *mass-independent*. Both these results are *opposite to the standard Migdal-Eliashberg theory*, where the self-energy slope is mass-independent and $T_{c0}$ is mass-dependent.

The *quasiparticle and replica bands* at $T=0$ are obtained from $\omega-\Sigma_{epi}(\omega)=0$. In the following we make calculations at $T=0$ and at the Fermi surface $\xi(\mathbf{k}_{F})=0$. The solutions are: ($1$) $\omega_{1}=0$ - the *quasiparticle band*; ($2$) $\omega
_{2}=-\Omega\sqrt{1+\lambda_{m}}$ is the $ARPES$ *replica band*; ($3$) the inverse $ARPES$ replica band $\omega_{3}=\Omega\sqrt{1+\lambda_{m}}$. The single-particle spectral function is $A(\mathbf{k}_{F},\omega,T=0)=\sum
_{i=1}^{3}(A_{i}/\pi)\delta(\omega-\omega_{i})$, where $A_{i}/\pi$ 