I saw the answer sometimes ago in a book "Quantum Electronics" or similar title. I don't remember the author since I lost the book. The book sets ( I believe so ) a constant electric current $I$ in a cable along the $z$ axis. Then, after a long ( but not straightforward ) calculation it arrives to the result I mentioned above.
Edit: A possible analogous calculation that OP refered to, Klein-Gordon particles generated by a classical source: problem 4.1 in these notes(In fact a solution manual to Peskin and Schroeder). If this is not what OP refered to, please delete it.