I recently finished a paper in which I received valuable feedback from two professors (neither supervisors nor coauthors) at some specific points, which ultimately leaded to localized significant improvements. I highly appreciated it and of course I added explicit mentions in the acknowledgements section stating their suggestions.

My question now is: I hardly will see again these professors for a long time. I thought about sending an email to them commenting on their inclusion on the acknowledgements... But then I thought that it should be something understood as obvious, without too much interest for them or even spammy. So, **shall I communicate their inclusion in the acknowledgements section? If so, what will be the correct etiquette for such an email?**

EDIT: As @StephanKolassa comments, another important and highly related issue is **what is the best timing for communication**, if this is considered adequate:

* First submission
* Acceptance
* Publication (early access or traditional)