WASHINGTON >> President Barack Obama’s compromise with Republicans on extending tax cuts for the wealthy, which his self-described progressive critics see as a profound betrayal, is bound to intensify a debate that has been bubbling up on liberal blogs and e-mail lists in recent weeks — whether the president who embodied “hope and change” in 2008 should face a primary challenge in 2012.

The idea seems to have little momentum for now, not least because there isn’t an obvious candidate, and because such a challenge would seem to have about as much chance of success as, say, a reality show about David Hasselhoff. That a primary is being openly discussed, though, reflects how fully Obama’s relationship with his party’s liberal activists has ruptured and the considerable confusion on the left over what to do about it.

Just last weekend, three liberal writers made the case for taking on Obama in 2012. Michael Lerner, longtime editor of Tikkun magazine, argued in the Washington Post that a primary represented a “real way to save the Obama presidency” in that it would force Obama to move leftward. Robert Kuttner, co-founder of the American Prospect and one of the party’s most scathing populist voices, issued a similar call on the Huffington Post, suggesting Iowa as the ideal incubator.

On the same site, Clarence B. Jones, a one-time confidante of the Rev. Dr. Martin Luther King Jr., suggested that liberals should break with Obama now, just as King and others did with Lyndon B. Johnson in 1968.

“It is not easy to consider challenging the first African-American to be elected president of the United States,” Jones wrote. “But, regrettably, I believe the time has come to do this.”

Meanwhile, in Iowa, a group known as the Progressive Change Campaign Committee, originally founded to aid Democratic congressional candidates in 2010, has started broadcasting an advertisement that shows Obama, in 2008, promising to reverse the tax cuts for the most affluent Americans. The group isn’t advocating a primary challenge just yet — but then, the choice of Iowa as a market seems intended to send a pretty clear warning to the White House.

“On issue after issue, when the public is on his side, this president just refuses to fight,” says Adam Green, the group’s co-founder. “At this point, the strategy is to shame him into fighting.”

All of this would have seemed unthinkable in 2008, when Obama’s red-white-and-blue visage seemed omnipresent on campuses and along city streets, a symbol to many of liberalism reborn. That, of course, was before the abandonment of “card-check” legislation for unions and of the so-called public option in health care, the escalation in Afghanistan and the formation of the deficit-reduction commission.

After this week’s reversal, quips progressive commentator Cenk Uygur, it may be time for his fellow progressives to face that Obama “is just not that into you.”

Of course, Obama is only the latest in a long line of Democratic presidents, going back to Franklin D. Roosevelt, to disappoint the liberal wing of his party and to at least hear rumblings of a challenge. In 1960, John F. Kennedy represented for liberals something similar to what Obama embodied as a candidate; two years later, the writer Norman Mailer acidly concluded that Kennedy stood for nothing but the pursuit of power, “without light or principle.”

Both Johnson and Jimmy Carter faced liberal primary challenges when they stood for re-election: Johnson because of the Vietnam War and Carter because he was deemed to be ineffectual in advancing liberal ideals. Bill Clinton’s stances on issues like free trade and welfare reform similarly infuriated the left, although he managed to avoid a primary fight.

Echoing his Democratic predecessors, Obama seemed frustrated at a news conference Tuesday about being pilloried by liberals who haven’t had to wrestle with the realities of governing.

“I’ve got a whole bunch of lines in the sand,” Obama protested.

The White House seems to view the notion of a serious primary challenge as far-fetched, and you can see why. For one thing, there seems to be no perfect vehicle out there, no Edward M. Kennedy biding his time.

The closest approximation appears to be Howard Dean, the former presidential candidate and party chairman who criticized the president’s deal on taxes. But Dean hasn’t shown any interest to this point in running, and you might recall that his 2004 campaign, for all its passion and fundraising prowess, yielded just two primary victories, in the District of Columbia and in his home state of Vermont.

There’s also the unique nature of this president himself, which makes the sheer math of any primary effort seem especially daunting. Obama, after all, drew his most monolithic support in 2008 from blacks and younger voters, two groups that are pivotal in Democratic primaries and whom you would expect to be essential constituencies for any kind of insurgent, take-it-to-the-Man candidacy.  

All that said, Obama must be aware that not all primaries of sitting presidents are about winning. Some, like Edward Kennedy’s in 1980 and Ronald Reagan’s in 1976, are designed to unseat the incumbent and capture the presidency. But other ideological challengers, like Eugene J. McCarthy in 1968 and Patrick J. Buchanan 20 years later, measure their success not by where they’re standing on Inauguration Day but by whether they have changed the trajectory of their parties.

Such protests that candidates don’t have to win more than a state or two to have an impact; they merely have to show up and sow division. It probably isn’t coincidental that none of the last four U.S. presidents to face primaries while seeking re-election — Johnson, Gerald R. Ford, Carter and George H.W. Bush — survived to serve a second term.

In other words, should the president’s progressive critics warm to the idea, it might not take a particularly credible primary challenge to weaken Obama’s chances for re-election. It might only take a challenge designed to do exactly that.

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