Plants go to a lot of trouble attracting ants not only through the nectar and pollen but through special organs high in the tree called nectaries. The sap sucking insects like scale and aphids only make it look worse, that is the source of your sooty mould, they attract the ants as well as the tree does.
In my opinion, I would be going after the sap sucking insects and leaving the ants to do their thing, they are great pollinators, I actually encourage ants into my greenhouse because they are the only pollinator I can get in there.
A very effective and low toxic way of controlling the sap suckers is as simple as a teaspoon of olive oil with a drop of dishwashing detergent to make it stick, into a 500ml spray bottle and away you go. Sap suckers breath through their skin, the oil blocks the pores so they suffocate.