Music has all kinds of crazy rights around it, there is the music itself, there are broadcast rights, performance licenses and so forth, a piece of music can have multiple rights holders…and from what I have seen, can be confusing as heck.
In terms of whether a campaign can use music or not, it depends. First of all, they would need to obtain a license to use the music and play it (assuming here that the politicians are likely playing a recording of the original song, which would involve I believe what they call mechanical reproduction rights(anyone familiar with that on here?),rather than performing it. It depends on who holds the rights and what kind of contract they have from what musician friends of mine have told me with the licensing agencies. So if BMI or ASCAP grants a license to use the music and play recordings of it, and the artist didn’t specifically have the right of refusal put into the contract with the agency, then the poitician could use it. Use in political campaigns does not have a broad waver from licensing, technically no one does, even groups like the Girl Scouts ran into problems with licensing agencies for songs sung around campfires (which quite honestly, was pathetic, it made the licensing firms look like greedy, ugly people, I believe they backed down when they realized how badly it made them look).
When conflicts have happened with artists and politicians, a lot of the time the politicians had not gotten licenses in the first place, they simply assumed they could use it, and if the artist disagreed with the politician it made it easy to get them to cease and desist. In other cases, the politicians had gotten the rights, but when the artist objected, rather than getting into a public fight over it, decided it was easier not to use it (there is sort of a parallel to this, parody is one of the few forms of music that does not require the person doing it to get a license/permission, but Weird Al for example makes it a point to get permission when he uses other artists music, he said he respects their feelings about how they want their music used. Funniest one had to be Paul McCartney refused him the right to use “Live and Let die” As “Chicken Pot Pie”, because Paul is a vegetarian:). Among other things, they simply may not want the ‘negative endorsement’ of a popular entertainer who objects to them using the music.