Two days after taking veiled jabs at his rival, Senator Ted Cruz took off the gloves and called out Senator Marco Rubio by name Thursday for his role in helping to author a 2013 immigration bill that would have provided immigrants in the U.S. illegally with a potential path to citizenship.

Cruz’s comments, on a program with conservative radio host Laura Ingraham, set off a war of words between the two senators, both children of immigrants, with each arguing the other was soft on illegal immigration. And that is raising concerns among some party strategists that the high-profile fight could further alienate Latino and Asian-American voters, wrecking the party’s chances in a general election where 30 percent of the electorate is projected to be non-white.

“This is disaster on all kinds of different levels,” said John Feehery, a veteran Republican strategist and lobbyist. “I’ve always been concerned that if we don’t get immigration right we have no chance to win this. And right now it doesn’t look like we are getting it right or we’re going to get it right.”