WASHINGTON — While a number of Supreme Court justices appeared troubled by a West Virginia Supreme Court justice’s refusal to recuse himself from a case involving his major campaign contributor, they struggled during arguments on Tuesday with what standard might be used to determine when due process requires recusal because of the appearance of bias.

Veteran Supreme Court advocates Theodore Olson, a partner at Gibson, Dunn & Crutcher, and Andrew Frey, a partner at Mayer Brown, faced off in Caperton v. Massey Coal, No. 08-22.

Olson urged the justices to hold that the Constitution can require recusal when there is the appearance or probability of bias, and the test is whether a reasonable person faced with all the facts would believe that the judge is biased. But Frey said an appearance of bias based on a “debt of gratitude,” as in this case, would never cross a constitutional line and was a “limitless” standard. The issue, he argued, should be left to state legislatures and judicial canons of ethics.

Hugh Caperton won a $50 million verdict against Massey Coal, owned by Don Blankenship, because of the way in which Blankenship drove Caperton’s mining company out of business.

Between the jury verdict and Blankenship’s anticipated appeal to the West Virginia Supreme Court of Appeals, there was an election for a seat on that court. Blankenship contributed $3 million in direct and indirect contributions to then-attorney Brent Benjamin in an effort to defeat the incumbent judge. That amount was 60% more than the total spent to support Benjamin’s campaign.

Benjamin won the election. When Blankenship’s appeal came before the court, Caperton asked Benjamin to step aside from the case because of the contribution connection, but Benjamin refused. Benjamin subsequently voted in a 3-2 decision in favor of Blankenship.

Chief Justice John G. Roberts Jr. called “probability” in probability of bias “a loose term,” and pressed Olson on what it meant: what percentage, more than 50%, he asked.

Olson responded that the court is familiar with the term and noted the use of probable cause in the Fourth Amendment context. “The court often decides the meaning not on the basis of mathematical certainty,” he said, adding that the reasonable person standard is also a familiar test.

Olson faced a very skeptical Justice Antonin Scalia who said, “We can’t run a system on such a vague standard. You give us nothing to hang onto but case-by-case circumstances.”

Olson acknowledged, “This is not easy,” but, he added, “The Conference of Chief Justices — they are the judges who would have to live with it — they said, `We need it.’ “

Justice Samuel A. Alito Jr. noted that Olson had suggested a five-factor test for when campaign contributions create the appearance of bias; the Conference of Chief Justices proposed seven factors; the American Bar Association, four factors, and Public Citizen, 10.

He and others put Olson through a series of hypotheticals as to whether recusal was required: when members of a trade association that had contributed to a judge’s campaign had cases before the judge or when the United Mine Workers, who made a get-out-the-vote effort on behalf of a judicial candidate, had cases before the judge.

Olson argued that not all appearances of bias gave rise to constitutionally required recusal. It depended on all of the circumstances, he said.

When asked if the political process was dealing with this issue, Olson said, “The political process is spiraling out of control, hasn’t done the job and the trend is running in the opposite direction.”

Frey just as steadfastly argued that the court has held that due process requires recusal in only two circumstances: when a judge has a financial interest in the case and in contempt proceedings, which have special rules.

“This is a situation where the states are dealing with it legislatively,” said Frey, noting that West Virginia, in response to the Benjamin situation, closed a contribution loophole for so-called 527 organizations. He also said almost every state had a standard governing the appearance of bias.

But Justice David H, Souter said, “It’s difficult to make the argument that the standard worked in this case.”

Justice John Paul Stevens appeared surprised when Frey said an appearance of bias was unlikely to ever give rise to a constitutional issue.

“We have never faced a case before this extreme,” said Stevens.

Justice Anthony M. Kennedy, who earlier had questioned the vagueness of Olson’s standard, told Frey, “It does seem to me the appearance of bias standard has much to recommend it.” He added that it does not require an inquiry into actual bias.

But Frey warned that an appearance standard based on “debt of gratitude” had “no reasonable limit.” He pointed to newspaper endorsements, contributions by doctors and lawyers, and union get-out-the-vote campaigns — all of which could be argued trigger a debt of gratitude.

A decision is expected by July.