Moment Generating Function of the Product of Two Independent Zero-Mean Normal Random Variables

Question

I want to find the moment generating function of the product of two independent random variables: $X \sim \mathcal{N}\left(0, \sigma^2 \right)$ and $Y \sim \mathcal{N}\left(0, \sigma^2 \right)$.

I think I have found the characteristic function:

$$ \varphi(t) = \sqrt{\frac{1}{1+ \sigma^4 t^2}} $$

Does this mean that the moment generating function is:

$$ M(t) = \varphi(-jt) = \sqrt{\frac{1}{1- \sigma^4 t^2}}$$

Or does it actually not exist?

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Questions:

1. Am I correct about the characteristic function?
2. Is that, therefore, the moment generating function?
3. If you find a characteristic function, how can you tell the moment generating function exists or does not exist?

Sorry for such a basic question but I am learning all of this on my own.

Thanks!