d_hat() is an estimated distance, while d() is the true distance. The paper assumes that there is a small number of labeled examples and a large number of unlabeled examples, So, when we use the labeled examples we are calculating an estimated distance (d_hat()). When we use the unlabeled examples we have a true distance (d()) because we are using lots of data points to compute the distance (although, in my opinion, this is still an estimate).