I'm wondering what exactly happens if a QFT is regularized with a smooth cutoff, but it is not sent to infinity at the end of the day. I'm certainly thinking about string theory's exponential smoothness around the string scale.
Polchinski says in his introduction to String Theroy vol. 1
In quantum field theory it is not easy to smear out interactions in a way that preserves the consistency of the theory. We know that Lorentz invariance holds to very good approximation, and this means that if we spread the interaction in space we spread it in time as well, with consequent loss of causality or unitarity.
The first thing that came to my mind was the Ostrogradski instability, but the theorem applies to a finite number of higher derivatives. A smooth cutoff however needs an infinite number of them. Another possible issue that I thought of is that the cutoff is normally enforced in Euclidean signature. But this affects loop momenta only, which are being integrated over.
I also found this blog post (paragraph about discretized theories) by Jacques Distler saying only that it requires fine-tuning due the cosmological constant.
So how exactly does a smooth finite cutoff violates unitarity?