The House Authorized Its Impeachment Inquiry Against Biden. Now What?

House Republicans this week formalized their impeachment inquiry into President Biden with a party-line vote, as leaders of the investigation argued they needed to take the step to ward off potential legal challenges to their work.

But they have so far found no evidence that Mr. Biden committed high crimes and misdemeanors, the constitutional standard for impeachment and removal.

Here’s what happens next.

Republicans are hunting for evidence of bribery, misconduct and obstruction.

Republicans are investigating three potential offenses they say they suspect Mr. Biden of having committed: bribery, misconduct in office and obstruction.

All three relate to the conduct of his son Hunter, who is under indictment on federal tax and gun charges. Hunter Biden received millions of dollars through his business dealings in Ukraine, China and other countries. A Yale-trained lawyer, Hunter Biden says that, even in the throes of addiction, he did legitimate work to receive the money, such as providing legal services, serving on corporate boards and working on financial and energy deals with overseas partners.

Republicans charge that he was merely peddling influence and selling access — or at least the perception of access — to his father, who served as vice president from 2009 to 2017.

But Republicans have yet to find evidence tying Hunter Biden’s business practices — including his alleged financial misdeeds — to his father.

To prove President Biden is guilty of bribery or misconduct in office, Republicans would most likely need to demonstrate that Mr. Biden either accepted money from or changed U.S. policy to benefit the business partners of his son. To date, they have proved neither.

Republicans have also alleged there was political interference — during both the Trump and Biden administrations — into the Justice Department’s investigation into Hunter Biden, who now faces 17 years in prison. But the interference allegations from two I.R.S. investigators, who have testified on Capitol Hill under whistle-blower protections, have centered on lower-level Justice Department employees, not President Biden himself.

Impeachment is a political process, so Republicans don’t actually need proof of anything.

The Constitution sets the standard for impeachment as commission of “treason, bribery, or other high crimes and misdemeanors.”

Functionally, though, there is nothing stopping the House — a political body, not a court of law — from voting to impeach Mr. Biden or anyone, even in the absence of strong evidence that they have done any of that.

“An impeachable offense,” former President Gerald R. Ford once famously said, “is whatever a majority of the House of Representatives considers it to be at a given moment in history.”

Former President Donald J. Trump has urged on House Republicans at every step to do to Mr. Biden what Democrats did to him twice, but some G.O.P. lawmakers have said they do not see a case for impeaching Mr. Biden yet.

That is unlikely to dissuade the Republicans leading the inquiry, including some of Mr. Trump’s closest allies in Congress, from continuing their impeachment push well into 2024, keeping Mr. Biden under an investigative cloud as the presidential campaign intensifies.

The G.O.P. has tried, so far without success, to tie President Biden to his son’s business.

Using their subpoena power, Republicans have obtained more than 36,000 pages of bank records; 2,000 pages of suspicious activity reports from the Treasury Department; and dozens of hours of testimony from two of Hunter Biden’s business partners, a senior official from the National Archives and Records Administration, seven federal agents and three U.S. attorneys.

The closest they’ve come to directly tying Hunter Biden’s business activities to his father is the testimony of his former business partner Devon Archer, who told investigators he could recall about 20 times when he and the younger Mr. Biden were meeting with business associates and Hunter Biden put his father on speakerphone. But Mr. Archer also said those conversations were only about niceties — “How’s the weather? How’s the fishing?” — and he said he was “not aware” of any wrongdoing by the elder Mr. Biden.

Republicans have also toiled, so far without success, to prove that Mr. Biden was enriched by his son’s business dealings.

They point to an unverified allegation the F.B.I. received that claimed Mr. Biden, while vice president, and his son had each accepted a $5 million bribe from an official with the Ukrainian energy company Burisma, on whose board Hunter Biden sat. That allegation has since been undercut by other evidence, including by the Ukrainian businessman who is alleged to have been involved and said that no such bribe occurred. Bank records obtained by House Republicans also have not shown evidence of a $5 million bribe.

In fact, many of the documents Republicans have produced thus far have demonstrated that Hunter Biden’s deals did not enrich his father. Instead, they show that the elder Mr. Biden lent money to his son and brother, James, when they were in need, and they later paid him back.

For instance, the House Oversight Committee released documents that showed that one of Hunter Biden’s businesses, Owasco PC, made three payments of $1,380 to the elder Mr. Biden in 2018, when he was not in office. Republicans said the payments were evidence of corruption. Other documents indicate the money was to pay back his father for helping to cover the cost of a Ford truck.

Republicans are looking for proof that Mr. Biden took corrupt official actions.

Republicans have also been unable to find proof that as vice president, Mr. Biden tried to change policy or otherwise use his power to benefit his son’s business partners. They have alleged that Mr. Biden’s push for the firing of a Ukrainian prosecutor who was investigating Burisma was a corrupt move meant to benefit his family, but that was the official position of both the U.S. government and the European Union at the time.

They have had more success in showing that the elder Mr. Biden has at times made false or misleading statements about his son’s businesses. For instance, he once claimed Hunter Biden had not made money from China when, in fact, he did.

Representative Jim Jordan, Republican of Ohio and the chairman of the Judiciary Committee, has suggested that any false statements the president made about his son’s business interests could constitute obstruction.

Authorizing the inquiry gave House Republicans more legal and political cover.

The White House essentially dared Republicans to call the vote by dismissing their investigation as illegitimate without the imprimatur of the full House. Richard Sauber, a special counsel to Mr. Biden, condemned the inquiry in a Nov. 17 letter and resisted making certain Biden administration employees available for closed-door interviews.

“You also claim the mantle of an ‘impeachment inquiry’ knowing full well that the Constitution requires that the full House authorize an impeachment inquiry before a committee may utilize compulsory process pursuant to the impeachment power — a step the Republican House majority has so far refused to take,” he wrote.

Republican leaders seized on the stance to persuade impeachment skeptics in their ranks that a formal investigation was the only way to overcome White House stonewalling.

The G.O.P. doesn’t have the votes to impeach Mr. Biden yet, and is unlikely to be able to remove him.

Mr. Jordan has laid out his plan to call in nine more witnesses for questioning within two months. In particular, Republicans are trying to force two tax investigators to testify about why Hunter Biden was not charged with felonies earlier.

Still, House Republicans don’t yet have the votes to move forward with impeachment. Several mainstream Republicans said in interviews that while they supported the investigation, they did not believe the evidence was there to impeach Mr. Biden.

Even if the House does so, the likelihood of removing him from office is virtually zero. The Senate is controlled by Democrats, who would be all but certain to lead the charge to acquit him.