The public deserves better than the she said, he said swearing match staged by the Senate Judiciary Committee on Thursday. In measured, dignified testimony, Dr. Christine Blasey Ford leveled an accusation of sexual assault against Supreme Court nominee Brett Kavanaugh. Judge Kavanaugh angrily, emphatically rejected it.

To anyone with any respect for the norms of criminal factfinding, this process was an embarrassment. If an accused’s indignant denials could short-circuit every serious criminal investigation, our justice system would be a wreck.

There’s no reason for this she said, he said contest when important witnesses remain unheard and lines of inquiry unexplored. So it’s a relief the FBI now will have the chance to pursue obvious leads and tie up troubling loose ends.

There is first of all Mark Judge, who Dr. Blasey said witnessed the young Brett Kavanaugh’s attack on her and at times egged it on. Her strongest memory of the event, she said, was “the uproarious laughter between the two, and their having fun at my expense.”

Over and over during the day’s testimony, one or another Republican senator and Judge Kavanaugh himself insisted Mr. Judge had submitted a statement “under penalty of felony” denying all knowledge of the alleged attack. Only later did we learn that Mr. Judge’s lawyer, not Mr. Judge, had signed this six-sentence statement.

No doubt Mr. Judge, who has made no secret of his youthful drinking, would prefer to avoid the spectacle of a public hearing. But where the integrity of the Supreme Court nominating process hangs in the balance, Mr. Judge’s desire to avoid questioning under oath must yield. Accusations of sexual assault often reduce to she said, he said credibility contests because typically only two persons were present at the crime scene. Here a third person was said to be in the room, yet he’s allowed to escape testifying.

Dr. Blasey named two other witnesses present on the day of the attack, though not eyewitnesses to it. They too have submitted statements “under penalty of felony,” apparently claiming no memory of the gathering Dr. Blasey described. Perhaps that will prove to be the sum of their knowledge. Yet closer questioning is in order, especially as one of these persons, Patrick “P.J.” Smyth, appeared again in some of the day’s most potentially revealing testimony.

That testimony concerned Judge Kavanaugh’s calendar entry for July 1, 1982. Judge Kavanaugh insisted in his opening statement to the committee that he could not have been present at the time of claimed attack on Dr. Blasey because the event she described, involving a few friends gathering for drinks in the summer of 1982, “presumably happened on a weekend.” He said his calendars from that summer “show that I was out of town almost every weekend night before football training camp started in late August.”

Here Rachel Mitchell, the Arizona prosecutor serving as staff counsel to the Republican members of the committee, spotted a glaring discrepancy. “I would like you to look at the July 1st entry,” she said to Judge Kavanaugh, referring to a Thursday. The entry reads, “Go to Timmy’s for Skis w/ Judge, Tom, PJ, Bernie, Squi.” Skis, we later learned, stands for brewskis, Judge for Mark Judge, and PJ for Patrick Smyth—three elements present at the scene alleged by Dr. Blasey.

Not long after Ms. Mitchell exposed this inconsistency in Judge Kavanaugh’s testimony, Republican senators reasserted control and airbrushed her from the scene like an apparatchik from a Kremlin photograph. We now must await the FBI’s investigation to learn where this line of inquiry might have led.

I don’t know where the truth of these events lies. I do know Dr. Blasey has staked her good name and peace of mind and the safety of her family on the truth of her allegations, and Judge Kavanaugh has staked his name and his family’s dignity on his denial. Resolving this conflict requires far more than a single chaotic day of piecemeal testimony and senatorial posturing.

We all deserve better, as we must live for decades with the consequences of this process.

George Fisher is the Judge John Crown Professor of Law at Stanford Law School and a former prosecutor.