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Mr. Brown released a radio spot last week commemorating the 100th anniversary of Fenway Park and embracing the Boston Red Sox, even though local media quickly pointed out that he pushed to move the team from the historic park in 2000.

What could tip the scales is how much outside groups pump in, analysts say.

Mr. Brown and Mrs. Warren struck a deal in January calling on third-party groups to stay out of the race. They agreed that if a group runs an ad, the benefitting campaign must make a charitable contribution worth half the ad’s costs within three days.

But that’s no guarantee that super PACs won’t jump in at some point. Before the deal, groups such as the Republican-affiliated Crossroads GPS and the liberal League of Conservation Voters had already poured millions into attack ads.

“I think they will,” said Deborah Schildkraut, a political science professor at Tufts University, saying there’s little chance super PACs will stay out of the marquee contest for long. “Frankly, there’s not much the candidates can probably do about it.”