QFTs with a finite cutoff 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?
EDIT: Today the following paper https://arxiv.org/abs/2111.03672 appeared analysing the non-local field theory effective action of the open string tachyon. As far as I can tell they do not discuss unitarity directly but show that by a field redefinition the theory is local in time and non-local in space only. This implies causality.
 A: One cheap way to see this maybe (as in: mathematical, but that doesn't connect immediately to Polchinki) is that a smooth cut-off in momentum space will necessarily alter the analytic structure of the answer and it's hard to see how this would not affect immediately the unitarity cuts. For instance, add a $\exp(-\ell^2 \alpha')$ to a scalar box integral, where $\ell$ is the loop momentum (one-loop) and $\alpha'$ some inverse mass scale. This shuts off high energy modes smoothly, but it stays in any unitarity cut as the exponential of some on-shell momenta and is not a function you expect from a tree-level graph in your theory, at least not if it's just conventional field theory with finitely many particles. Unitarity tells you that at tree-level you might have only poles. This exponential has a bad divergence at infinity on the complex plane. I'm not sure how this relates directly to Polchinski's "wrapping in time" argument though.
Also, note that in string theory, the exponential smoothness of the amplitudes is not a loop effect, it's the result of delicate cancellations between the infinite tower of states. These guys http://arxiv.org/abs/1607.04253 proved that it's universal to any theory of weakly interacting higher spin particles. In string theory, even at loop-level, you don't have a sharp cut-off in momentum space, it is in Schwinger proper-time space that the integration is truncated (as is explained somewhere in Polchinski).
