A crime? Doesn't really matter.
Section 6
... and for any Speech or Debate in either House, they shall not be questioned in any other Place.
The Constitution says the courts have no say...
The Constitution says the courts have no say...
Not so; that prohibition against questioning applies to "speech or debate" in one of the Houses -- this certainly does not mean that they cannot (a) be called as witnesses for some act, or (b) cannot be tried for crimes committed.