chase shadows

To pursue someone or something that remains elusive. We've followed up on every lead, but so far we're still chasing shadows. We've turned up no evidence, no suspects—nothing. I can't believe the research team is still chasing shadows. They're going to lose funding if they don't find a cure soon. A: "The girl disappeared after school last Tuesday, and we've been chasing shadows ever since." B: "How can this be? She didn't just vanish!"