between the Higgs doublet and $\eta$ are forbidden. The lightest field in the dark sector can play the role of WIMP dark matter candidate, that can be detected by nuclear recoil experiments such as XENON-1T [@Aprile:2018dbl]. The mass spectrum for the components of the $\eta$ doublet are given by $$\begin{aligned}
 m_{\eta^{R}}^{2}&=m_{\eta}^{2}+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}+2\lambda_{5}\right)\\
  m_{\eta^{I}}^{2}&=m_{\eta}^{2}+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}-2\lambda_{5}\right)\\
 m_{\eta^{+}}^{2}&=m_{\eta}^{2}+\frac{1}{2}\lambda_{3}v^{2}.\end{aligned}$$

![ []{data-label="Solar Loop Diagram"}](plots/SolarLoop.pdf){width="40.00000%"}

The total neutrino mass has the structure $$\begin{aligned}
 \mathcal{M}_{\nu \text{TOT}}^{ab}=-\frac{v^{2}}{M_{N}}Y_{\nu}^{a}Y_{\nu}^{b}+\mathcal{F}(m_{\eta^{R}},m_{\eta^{I}},M_{f}) M_{f}Y_{f}^{a}Y_{f}^{b},
 \label{mass matrix for scoto-seesaw}\end{aligned}$$ where the first term is the tree level neutrino mass that comes from the seesaw part, and the loop function $\mathcal{F}$ characterizes the quantum correction arising from Fig. \[Solar Loop Diagram\]. This is responsible for inducing the solar mass scale. The loop function $\mathcal{F}$ is expressed as the difference of two $B_{0}$-Veltman functions, namely, $$\begin{aligned}
 \mathcal{F}(m_{\eta^{R}},m_{\eta^{I}},M_{f})=\frac{1}{32\pi^{2}} \left(\frac{m_{\eta^{R}}^{2}\,\text{log}\left(M_{f}^{2}/m_{\eta^{R}}^{2}\right)}{M_{f}^{2}-m_{\eta^{R}}^{2}}-
 \frac{m_{\eta^{I}}^{2}\,\text{log}\left(M_{f}^{2}/m_{\eta^{I}}^{2}\right)}{M_{f}^{2}-m_{\eta^{I}}^{2}}\right)\end{aligned}$$ Both terms in Eq. \[mass matrix for scoto-seesaw\] have a projective nature. Hence one out of the three neutrinos remains massless. From the eigenvalues of the neutrino mass matrix $\mathcal{M}_{\nu}^{ab}$ one can estimate the solar and the atmospheric square mass differences as, $$\begin{aligned}
 \Delta m_{\text{ATM}}^{2}\sim \left(\frac{v^{2}}{M_{N}}\mathbb{Y}_{\nu}^{2}\right)^{2},\,\,\, 
 \Delta m_{\text{SOL}}^{2}\sim \left(\frac{1}{32\pi^{2}}\right)^{2}\left(\frac{2\lambda_{5}v^{2}}{M_{f}^{2}-m_{\eta^{R}}^{2}}M_{f}\mathbb{Y}_{f}^{2}\right)^{2}.
 \label{atm and sol mass}\end{aligned}$$ where we take $M_{f}^{2}$, $m_{\eta^{R}}^{2}$, $M_{f}^{2}-m_{\eta^{R}}^{2}\gg\lambda_{5}v^{2}$ and $\mathbb{Y}_{\ell}^{2}=(Y_{\ell}^{e})^{2}+(Y_{\ell}^{\mu})^{2}+(Y_{\ell}^{\tau})^{2}$ for $\ell=\nu,\,f$.\
From Eq. \[atm and sol mass\], it 