As a follow up to my previous question (http://mathoverflow.net/questions/168077/how-does-constructive-quantum-field-theory-work), I was wondering what difficulties physicists have had constructing 4d axiomatic qfts. Why has CQFT's success in 2 and 3d spaces not been extended to 4 dimensions? Once again, any level of answer is okay, but technical is preferable.