Bruce Ackerman and Oona Hathaway note that the current legal basis for the U.S. military operation in Iraq is the second prong of the 2002 AUMF which grants the president the authority to use the military to "enforce all relevant United Nations Security Council resolutions regarding Iraq."
But here's the rub. The most recent U.N. resolution expires on Dec. 31, and the administration has announced that it will not seek one for 2009. Instead, it is now negotiating a bilateral agreement with the Iraqi government to replace the U.N. mandate.
Whatever this agreement contains, it will not fill the legal vacuum. That's because the administration is not planning to submit this new agreement to Congress for its explicit approval. Since the Constitution gives the power to "declare war" to Congress, the president can't ignore the conditions imposed on him in 2002 without returning for a new grant of authority. He cannot substitute the consent of the Iraqi government for the consent of the U.S. Congress.
But of course Bush (and John McCain!) want a permanent American military presence in Iraq, but they know congress won't authorize such a presence. Hence, the only solution available to them is to ignore the law and the constitution and just keep the troops. Ackerman and Hathaway suggest merely continuing the U.N. resolution for another year, which will give the next president a legal basis for doing whatever he's been granted a mandate to do (since no matter who wins the troops can't be made to suddenly vanish in a puff of smoke in January). But, obviously, for that to happen the administration would need to concede that it lacks the legal and constitutional authority to ignore congress and they'll never do that.