Classification of Mukai pairs with dimension 4 and rank 2

We give the complete classification of Mukai pairs of dimension 4 and rank 2 with Picard number one, that is, pairs (X,\mathcal {E}) where X is a Fano 4-fold with Picard number one, and \mathcal {E} is an ample vector bundle of rank 2 on X with c_1(X) = c_1(\mathcal {E}). Equivalently, the present paper completes the classification of ruled Fano 5-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007.