Men who take paternity leave without doing at least half of the baby care.

That would be nearly all of them, if we are to believe this study:

Only three of 109 male faculty members surveyed reported that they did half or more of the care, while 70 of 73 women reported doing at least half. On average, both men and women professors reported that the mother did more than half the work for all 25 of the child care tasks. This result holds even when the male professor's wife works full-time.

Was breastfeeding included as one of the tasks? That would skew results. Also, these were professors. Professors are comfortable taking leave time away from work.

The female professors also reported higher average enjoyment scores than males on 24 of the 25 child care tasks. (The sole exception was managing the division of labor for parenting tasks, which men disliked less than women.)

First of all, maybe those women are maintaining their self-esteem by getting into the frame of mind where they think about themselves as loving what they are doing. Also, that management-oriented man might be managing her moods, taking care of her, and that might not have been counted as one of the "child care tasks."

Interestingly, the report suggests that paternity leave be eliminated, because men are using it to further their careers, thereby creating greater inequity for women who actually take time off.

Well, you can't do that. That would be illegal sex discrimination. But let's assume you could do it. Does it make sense? Let's give this advantage only to women, because, in the great majority of the cases, women will use it for the "right" reason and men will, as men tend to do, find ways to take selfish advantage of the opportunity. That's horrible sterotyping, which is why it's illegal, but quite aside from that, I don't like incentivizing the female professors' failure to take advantage of time off from teaching to work on their scholarship. And I don't like giving up on the ongoing project of mothers and fathers working out childcare arrangements together. And it's none of the employer's business how a man and a woman structure their activities within the home.