Do wood ants live alone

No. No true ant lives alone; they all live in colonies. The wingless wasps called velvet ants do live alone (we say that they are solitary) but they are not really ants.

The closest you can come to a real ant that lives alone is a species called Anergates atratulus. The queen invades the nest of other ants of a kind called Tetramorium caespitumand kills their queen. Their workers then look after the new queen and feed her till they die. She lays eggs that hatch into more Anergates queen larvae, but her eggs do not grow into workers, so when the Tetramorium workers die of old age, the Anergates queen dies too. So you could say that she lived alone, without her workers, but at least she did not, and could not, live without the other Tetramorium workers. I don't call that living alone.