Consider a massive scalar diagram such as
The loop momentum enters and exits the tadpole vertex, so that in the first diagram the momentum in the propagator connecting the two vertices is zero due to overall momentum conservation. This is ok if the fields are massive.
However, in the second diagram the propagator connecting the two vertices has exactly the same momentum as the rightmost external leg, and is therefore on-shell and blows up!
I know that tadpole loop momentum integration develops a divergence and is e.g. dimensionally regularized. But the non-loop propagator being on-shell simply makes the result infinite regardless of dimensional regularization!
How to make sense of this?