In root locus technique, which branch emerging from the open loop poles goes up and which goes down?
There is no such factor. 

The root locus tracks the closed loop pole locations from the open loop poles. As the gain K is increased little by little, it is easy to see how they move towards the breakaway point. Another way to put this is, each time the gain is incremented, the poles move a little and it could be assumed that the new pole location that is very close to the old pole location must be related. 

The problem is that when the poles reach the break out point they are real and repeated. When both poles are the same, mathematically they are indistinguishable. Because they are the same, there is no way to know which one will go up and which one will go down because you don't know which pole is which.  

Think of it like putting two identical marbles into a black box. When you take them back out, how do you know which one is which?

The computer program arbitrarily chooses which branch goes down and which branch goes up to make the plot look nice. There is no particular logic to it.