From a Constitutional perspective, anything the House wants and two thirds of the Senate will go along with.
In keeping with the spirit of the Constitution, judges must be allowed to keep their positions as long as they exercise "good behavior". It would not be unreasonable under the spirit of the Constitution to regard a violation of one's oath of office as "bad behavior".
To be sure, reckless impeachment of judges would make the SCOTUS even more overtly political than it already is, but it might also help wake up the public to something very important: the fact that what the Constitution means, and what a well-behaved court says it means, will be one and the same, does not imply that the Constitution always means what the court says it means. The Court has no actual authority over anyone other than the actual parties to the cases brought before it. If the Court overturns a ruling by a lower court that says a statute in unconstitutional, that doesn't mean the statute is constitutional. Rather, it conveys a message that any future lower court rulings which say the statute is unconstitutional are likely to be overturned, and that judges who don't want their rulings overturned should avoid issuing rulings that will be. That's it.
If five judges say one thing and four say another, the ruling by the five judges will be binding upon the actual parties to the case. On the other hand, if the ruling by the four judges cites reasonable constitutional justification and the ruling by the five judges does not, those who are sworn to uphold the Constitution should recognize that Supreme Court edicts are not superior to the Supreme Law of the Land.
Thanks, that helps. Gives me cause to do more research.