When does a political dirty trick become a crime? When does an investigation of one become political lawfare, the use of law enforcement to harm or intimidate one’s political opponents? The U.S. attorney for Alaska said last week that federal law-enforcement agencies have been investigating the “two Sullivans” on the Alaska ballot for U.S. Senate.

Dan J. Sullivan, who has no political experience, filed as a challenger to the Republican incumbent, also named Dan Sullivan. Both Dan Sullivans made it past the first round in Alaska’s byzantine ranked-choice voting system and will be among four candidates on Tuesday’s ballot. This was widely suspected to be an effort to deceive voters at Sen. Sullivan’s expense and to the benefit of the only Democrat on this week’s ballot, former Rep. Mary Peltola.

Last week the New York Times confirmed those suspicions, reporting that “the Peltola campaign’s lawyers, as well as senior aides at the Democratic Senatorial Campaign Committee,” worked “with people trying to recruit Mr. Sullivan to run for Senate.”“Politics ain’t beanbag,” as Finley Peter Dunne’s fictional Chicago saloonkeeper, Mr. Dooley, famously observed.

But throughout American history, political combat has remained subject to legal limits. The boundary between shady and illegal activity isn’t always clear, but there is a zone that, when entered, can expose political operations and operatives to accusations of criminal conduct in their political affairs. One way the dirty trick can easily become a criminal matter is through misconduct by the participants in the course of investigation of a dirty trick. Watergate is the most prominent example. The dirty trick was itself a crime—a burglary seeking kompromat of the opposition party by bugging its headquarters.

The men who carried it out were charged with burglary, wiretapping and conspiracy. But it was the attempts by its intended beneficiaries, including President Richard Nixon, to conceal the crime and hinder the investigation for which several appointees, aides and operatives went to prison and Nixon resigned rather than face impeachment. Thus did that dirty trick get turned on its head, producing the adage “The coverup is worse than the crime.”In the Alaska matter, an intentional effort to confuse voters to harm Sen. Sullivan’s re-election surely falls under the rubric of dirty tricks, which Merriam-Webster defines as “underhanded stratagems for obtaining secret information about or sabotaging an enemy or for discrediting an opponent (as in politics).”

But is it a crime—or could it become one by way of conduct during an investigation? Perhaps the Times got it wrong, and Dan J. Sullivan simply woke up one morning and decided to run for the Senate. That wouldn’t be a crime, even if he did so to make the senator’s re-election less likely. But if political operatives counseled or cajoled him into running, a detailed inquiry into who, what, when, where and how is justified. The most politically salient question: What did Mrs. Peltola know, and when did she know it?

Mrs. Peltola has given her answer. She told Anchorage’s KTUU-TV that she had “no idea.”If that isn’t true, the worst consequence of lying to the press is embarrassment and perhaps ultimate repudiation at the ballot box. But the false denial of even a small detail in a federal inquiry can be a felony.

If political operatives were behind the duplicate Sullivan’s candidacy in some fashion, they have every political reason to conceal that role. Likewise, if Mrs. Peltola did anything at all to encourage or facilitate the duplicate Mr. Sullivan’s running, she too has every political reason to conceal that conduct. Such denials would be the most likely source of criminal allegations in the matter if the allegations have factual legs.

But what suspected underlying crime might trigger an investigation? An intentional effort to obfuscate the candidate choices in the election could itself be criminal. At some level, not yet precisely defined by courts, candidates for federal office are entitled to an election process that affords them fundamental fairness under the Constitution and statutory law. Under Bush v. Gore (2000), those rights are guaranteed by the Equal Protection Clause of the 14th Amendment.

Under 18 U.S.C. 241, it is a crime for two or more persons to conspire to “injure, oppress, threaten or intimidate any person . . . in the free exercise of any right or privilege” provided by the Constitution. Just as the duplicate Mr. Sullivan had as much right to run for office as Sen. Sullivan, the latter enjoys the right to run without a political cabal injuring or oppressing his right to do so through fraud and misrepresentation.

While courts may struggle in discerning the parameters of a fair election process, under this statute the circumstances as reported thus far would justify an inquiry into reported covert insertion of a duplicate Mr. Sullivan in search of political advantage. That seems especially so in this election, in which ranked-choice voting magnifies the potential consequences of such deception.

More generally, the conduct the Times reported, if true, is fundamentally dishonest, and the law likes to find a way to hold frauds and cheats accountable. Moreover, law sweeps broadly by rendering liable anyone who “aids, abets, counsels, commands, induces or procures” another to commit a federal crime, and conspiracy, once shown to exist, ensnares any person who acts to further its objective. Not every political dirty trick is a crime. But given the broad reach of federal criminal law, conspiring to subvert a fair election process is likely a crime by some measure.

Unfortunately, the votes are likely to be counted before the facts are fully known. Yet an effort to find those facts will remain worthwhile, because political dirty tricks have a way of recurring. If this was a dirty trick and not merely Dan J. Sullivan’s lark, investigation of the facts can help draw lines that determine when underhanded manipulation of the voting public and the voting system is a crime. That is a worthy undertaking.

The need to know takes the investigation out of the realm of politicized lawfare and into a legitimate inquiry with a lawful purpose. For the same reasons, a failure to investigate would be a dereliction of duty.

Mr. Terwilliger is a Washington lawyer. He served as deputy U.S. attorney general, 1991-93. [video] WSJ Opinion: So, Who’s Going to Win the Midterms?