Will California Gov. Arnold Schwarzenegger (R) leverage his popularity on behalf of Republican Bill Jones, the former secretary of state who’s making an underdog bid to unseat two-term Democratic Sen. Barbara Boxer? For now, the signs are — at best — mixed.

Last week, Schwarzenegger helped the Golden State GOP raise $2 million at a gala fundraiser in Los Angeles — a harvest that will aid Jones, at least indirectly. But the governor has not yet shown that he will lend his celebrity — or his political capital in what is usually a Democratic-leaning state — to keep Jones’ water-treading candidacy afloat.

Jones generously credits Schwarzenegger’s endorsement before the March Republican primary with helping him take a modest lead in the polls and turn it into a rout on Election Day. Jones also makes it clear that he will seek to tie himself to the superstar governor’s coattails as much as possible.

“In all of my time in California, I’ve never seen anyone drive the numbers the way the governor can,” Jones told reporters in Washington, D.C., recently.

Schwarzenegger’s approval ratings continue to float in the stratosphere. A Field Poll released last week found that 65 percent of Californians approved of the governor’s performance — including 48 percent approval among Democrats.

Schwarzenegger’s overall approval ratings were the highest of any Golden State governor since George Deukmejian (R) chalked up similar numbers in the 1980s. It even surpassed the highest mark ever achieved by Ronald Reagan — the gold standard for popular Republican governors in California.

The same poll, however, showed Jones trailing Boxer by 23 points among likely voters. Recent internal Republican polls have put Boxer about 10 points ahead, which is where she was in the last Field poll published on the Senate race in February.

Perhaps most troublesome for Jones was that even though he was the last Republican before Schwarzenegger to win statewide office, he remains a mystery to most voters. Two-thirds of the Field Poll’s respondents did not know enough about him to form an opinion.

This is precisely where the governor can help — if he chooses. Although California Republicans expect Schwarzenegger to stump at least a bit for Jones, just how much he’ll do remains to be seen.

“We haven’t nailed anything down with the governor,” Jones spokeswoman Valerie Walston said last week. “But there are plenty of opportunities.”

Jones was one of just three candidates Schwarzenegger endorsed in the March GOP primaries. But the governor has yet to make public a full list of those he intends to help in the general election.

A D.C.-based Democratic operative with close ties to California politicians said, “I would think he’d want to help Jones quite a bit.”

But Todd Harris, a spokesman with Schwarzenegger’s political operation, said that as far as he knows, nothing has been decided about the governor’s role in the Senate race.

The Senate race isn’t the only question mark for Schwarzenegger. He is also the honorary chairman of President Bush’s re-election campaign in California and has yet to nail down exactly what role he will play in that race.

On Tuesday, Schwarzenegger was quoted in the California Journal political magazine saying that he does not plan to campaign for Bush outside the state.

A decision to go all out for Bush — or Jones — would not be cost-free for Schwarzenegger. Both Bush and Jones lag in all Golden State polls, and Schwarzenegger — who is muting his partisan edge in a so-far-successful bid to govern from the center — may not want to invest much political capital in underdogs.

“When the question arises about President Bush, I would tell people that I’m supporting him and I will support him, and if he comes out here I will support him. … But I do not go around the country or anything like that, because there’s too many Democrats here,” Schwarzenegger told California Journal. “I don’t want to rub that in their face. I want them and me to be a partner, to work together.”

Kevin Spillane, a Sacramento-based Republican consultant who worked for former U.S. Treasurer Rosario Marin — the runner-up to Jones in this year’s GOP Senate primary — said the governor has identified a few candidates for the state Legislature he intends to help in November. Of those, only one is running against a Democratic incumbent.

“He’s being very careful about husbanding his political capital,” Spillane said.

Another kind of capital — financial — is one that Jones needs plenty of. Through March 31, Boxer had $5.9 million in the bank; Jones, despite running almost no TV ads during the Republican primary, had a mere $216,000.

Jones did get some help from a celebrity politician last week when Senate Majority Leader Bill Frist (R-Tenn.) appeared with him at a fundraiser in San Diego. “The fact that you’ve got the majority leader here raising money … shows my belief in Bill Jones as a candidate and as your future U.S. Senator,” Frist said, the San Diego Union-Tribune reported.

Frist’s trip came one day after the California GOP fundraiser headlined by Schwarzenegger. California GOP leaders say that Schwarzenegger is the state party’s best fundraiser since Reagan was governor.

Encouraged by Schwarzenegger’s popularity and a recent uptick in GOP voter enrollment in California — combined with a desire to make California at least remotely competitive in the presidential race — the Republican National Committee recently sent $500,000 to the state party.

This transfer should help Jones indirectly. But even if Schwarzenegger goes to the mat for Jones, Democrats say it won’t be enough to topple Boxer.

“We’re not taking the California race for granted,” said Brad Woodhouse, a spokesman for the Democratic Senatorial Campaign Committee. “Barbara Boxer’s not taking the California race for granted. But I don’t think even Arnold Schwarzenegger when he won the Mr. Universe contest could help Bill Jones. Jones is out of step with the voters, out of touch and, the last I looked, out of money.”