Database
db={"B":[{"_id":"1",name:"lola"}],"A":[{_id:"1",relation:{ref:"B/1"}},{_id:"2",relation:{ref:"B/2"}}]}
Query
db.A.aggregate([{$project:{"relation.ref":{$arrayElemAt:[{$split:["$relation.ref","/"]},1]},a:"$$ROOT"}},{$lookup:{from:"B",localField:"relation.ref",foreignField:"_id",as:"b"}},{$match:{$expr:{$eq:[0,{$size:"$b"}]}}}])