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While reading about exotic decays of Higgs boson one of the simplest interaction that we come up with which leads to BSM decays is:

$$\Delta L = \frac{\zeta}{2}s^{2}|H|^{2}.$$

This is the interaction of a singlet scalar field $s$ through the Higgs portal. It is mentioned in this document arXiv:1312.4992 on page 10 that for simplicity it is assumed that $s$ has a conserved $\mathbb{Z}_2$ parity.

I know about the intrinsic parity of fundamental particles, but I have never come across $\mathbb{Z}_2$ parity. What does it mean?

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Generally, such a statement implies that there is a $\mathbb{Z}_2$ symmetry of the theory, i.e. a symmetry operation that yields the identity if applied twice. Then basically, fields can be "even" or "odd" under the $\mathbb{Z}_2$ action, menaing the flip sign or not, and the Lagrangean itself is even, i.e. invariant.

In the case you mention, it presumably means that under this $\mathbb{Z}_2$, the new field transforms as $s\mapsto-s$ and all the SM fields are even. Then the $\mathbb{Z}_2$ symmetry forbids any linear coupling $\mathcal{L}\supset s\cdot (\text{SM fields only})$.

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