In what circumstances will a witness be given formal immunity to testify in a murder trial?

In People v. Badgett, supra, 10 Cal.4th 330, the defendants were tried for murder. The primary prosecution witness was the 17-year-old girlfriend of one of the defendants, who testified he had admitted killing the victim to her. (Id. at pp. 339-340.) After the juvenile witness had been detained, her mother was advised by the authorities the witness would be released from custody if she cooperated with the police. (Id. at p. 354.) The mother conveyed this promise to her daughter, who then provided a statement to the police implicating the defendants. Subsequently, the witness was given formal immunity and testified pursuant to an immunity agreement. (Id. at p. 340.) The court addressed and rejected the defendants' argument that the witness's statement was involuntary because it was induced by a promise of leniency, specifically the promise she would be released from custody. (Id. at p. 355.)