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The type-I seesaw mechanism yields the effective neutrino mass $$m_\nu\sim -M_DM_N^{-1}M_D^T.$$ Here, $M_D$ is the Dirac neutrino mass coming from Yukawa coupling of the left-chiral neutrino with the standard model (SM) Higgs and $M_R$ is the mass of the right-chiral neutrino whose scale is not known a priori. Without fine-tuning the Yukawa couplings $Y$ in $M_D=Yv$ (where $v=246$ GeV, the VEV of SM higgs) one has $M_D\sim 100$ GeV. Now, if one requires a neutrino mass of the $0.1$ eV, then one must choose $M_N\sim 10^{12}$ GeV.

However, this paper, below eqn. 2.2, says that

However, theoretical arguments based on the naturalness of the SM Higgs mass of 125 GeV against radiative effects induced by the neutrino loop suggest the seesaw scale to be below $\sim 10^7$ GeV.

Is this scale of $M_N$ viable without fine-tuning the Yukawa couplings of $M_D$?

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Small neutrino masses from a rather small right-handed neutrino mass scale always imply small Yukawa couplings. Note, however that $M_D \sim 100$ GeV assumes Dirac neutrino masses of the order of the top quark, by far the largest Yukawa coupling in the Standard Model. If you assume that the neutrino Dirac masses are of the scale of the lepton Yukawa couplings, you obtain (for the heaviest state) $M_D \sim 1$ GeV, two orders of magnitude smaller than before.

Since the Dirac mass enters the Seesaw formula quadratically, this lowers the realistic right-handed mass scale by four orders of magnitude. This takes your $10^{12}$GeV down to $10^8$ GeV, very close to the limit you quote.

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