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Seesaw scenarios can be understood systematically by looking at the Weinberg operator. The Weinberg operator $\frac{1}{\Lambda}$ LL \phi \phi$ is a (non-renormalizable) dimension-5 operator that yields nonzero masses for the neutrinos. Since it is an dimension-5 operator, it is suppressed by some large mass scale $\Lambda$. The origin of the Weinberg operator can be understood in terms of concrete seesaw scenarios (UV-completions).
There are three seesaw scenarios that yield the Weinberg operator in the low-energy limit when we integrate out all heavy fields and thus use an effective field theory approach.
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Seesaw models are extensions of the Standard Model of particle physics that aim to explain why neutrino masses are so tiny.