Why did Jeanne kill Paul?

The movie doesn't provide an answer, we can only guess. For instance this discussion in Slant magazine tries to offer an explanation:

EH: It’s yet another moment where we’re left to wonder what Jeanne is thinking, and any of the motivations you suggest seem plausible for this enigmatic woman. If I had to choose, I’d say that Jeanne is most disturbed by Paul’s flippant appropriation of her father’s military cap. When Paul approaches her in that hat, it’s Jeanne’s opportunity to kill a father who, the film hints, seems to have had a lingering impact on her life and her sexual identity: think of her mother’s very funny line about the enduring erotic charge of her dead husband’s boots, a sexual electricity that has transcended death (“They give me strange shivers,” she says). As we’ve discussed, Jeanne’s psychology isn’t examined as intently as Paul’s, but it’s clear that for her, military power and fatherly authoritarianism and sexual pleasure are all tangled together. Certainly, killing fathers was very much on Bertolucci’s mind at this point. Two years earlier, in 1970, he’d released a twinned pair of films about the often unwelcome power of the father figure: The Conformist, in which Bertolucci cheekily gives an assassination target the real-life Paris address and phone number of the director’s cinematic father Jean-Luc Godard, and The Spider’s Stratagem, in which the mystery of a father’s long-ago death reshapes the life of his son.