I was wondering why the article didn't show the actual ansatz anywhere. So I looked it up on Wikipedia [1] and then I understood why. It's a product over pairs around a sum over permutations around an exponential of a sum over pairs involving a "scattering phase shift function" (and I'm simplifying here).
Still would have appreciated it being in the article.
[1]: https://en.wikipedia.org/wiki/Bethe_ansatz#Discussion