Put down that smartphone, stand very still, and listen for the soft whoosh of hundreds of Capitol Hill Republicans breathing a sigh of relief. Stay quiet a moment longer, and you may hear a few even begin to weep gently.

The cause of such collective release? The appointment of Robert Mueller as special counsel charged with determining if any members of Team Trump got too cozy with Russia during last year’s campaign—and whether further improprieties may have flowed from that coziness. The vast assortment of Trump haters may be more delighted to see the president placed in such an ungainly position. But it’s safe to say no one is more relieved by Mueller’s arrival than GOP lawmakers.

To some degree, of course, the out-of-control freak show that is the Trump White House reflects badly on the entire Republican brand. But, let’s face it: That was happening anyway, with GOP lawmakers trapped on the frontlines. (The halls of the Capitol were so jam-packed with frenzied reporters this week that the Senate press gallery had to alert media outlets that the building had “reached its capacity.”)

Mueller’s appointment by deputy attorney general Rod Rosenstein potentially improves these members’ lives in multiple ways. Some are more immediate and/or direct than others, and most will rely at least a wee bit on Trump’s not going totally berserk and perpetuating the non-stop toxic chaos of the past several days. (This is no sure thing.) But when it comes to the uneasy, ever-shifting balance of power between Trump and Congress, special counsel Mueller places a fat thumb on the scales in lawmakers’ favor—for now, at least.

The most obvious benefit for Hill Republicans: Much of the pressure is off for them to pass judgment, both officially and unofficially, on their president’s behavior.

Most immediately, Democrats can no longer use demands for a special counsel as a way to put Republicans in an awkward spot.

“Democrats had the advantage that they weren’t responsible for anything,” said Representative Tom Cole. They could file discharge petitions calling for a special prosecutor or an independent commission, he said, and if Republican members didn’t sign on, the minority would “hammer” them. Meanwhile, Republicans who did voice openness to an independent investigation risked being tarred as disloyal by members of their own party (most definitely including Trump himself). “They enjoyed us twisting in the wind,” said Cole of Democrats. “It’s all moot now.”

Better still, Mueller gives members an easy way to brush off the media hordes. All those salivating journalists sticking mics and recorders in lawmakers’ faces, demanding comment on this or that, can now be ever-so-politely referred to the special counsel’s office.

“The media has over-sensationalized some of this stuff,” said Cole. “This holds them in check. It’s something they can follow and report on.” The process should become more organized and less leaky, said Cole hopefully. “I don’t think the media will be able to penetrate this investigation to the same degree that they seem to have penetrated the institutions of government.”

As for the multiple House and Senate committees already looking into the Russia issue: They are now expected to gear down and let Mueller take the lead. (Criminal probes trump oversight hearings.) Bringing in a special counsel means Congress is “pretty well knocked out of the game,” Senator Lindsey Graham told the New York Times. “And that’s probably the way it should be.” Don’t expect too many Republican members to get irate about having their authority usurped. Generally speaking, probing a president from one’s own party is a political nightmare that only ends in tears.

And in the event that Mueller does uncover something damaging to the White House? At least Congress won’t bear the blame for his appointment. “It wasn’t forced on [Trump] by a Republican House and Senate,” said Cole. “In that sense, hopefully there will be no bitterness directed at our leadership, let alone members. By and large, we stayed with the president.”

Beyond the particulars of this investigation, many Hill Republicans are hopeful that, with the white hot spotlight now shifting to Mueller, they can focus less on what Senate Majority Leader Mitch McConnell drolly referred to as the White House’s “drama” and more on, say, passing legislation.

“This actually allows us to get back to work, since this whole thing will be in Mueller’s hands,” a Senate GOP staffer told me.  “I think [Trump’s] crazy shenanigans have wasted at least a month of the congressional schedule if not more. Wasted,” the staffer repeated. “And I think people are incredibly frustrated and angry that he is throwing away this once-in-a-generation opportunity for conservatives to have some real accomplishments because the president is letting this petty stuff distract to the point of paralysis.”

“It is like a weight has been lifted,” cheered another Senate Republican aide. “We don’t have to pay attention to the latest Trump news anymore. It’s all Mueller’s problem now.”

Indeed, arguably never have congressional leaders been better positioned to ignore Trump. The Russia heat can be directed elsewhere. Meanwhile, with Trump’s attention (and potential fury) focused on the special counsel’s probe, he has less time, energy, and Twitter characters to devote to making McConnell’s or Paul Ryan’s lives miserable by criticizing their slowness or effectiveness or pushing for unpalatable policies like border walls and costly infrastructure plans. “This provides everyone with some breathing room,” said the staffer.   

Not that anyone is breathing too easy. Where things go from here depends heavily on Trump, acknowledged the staffer. “If he is disciplined enough to also get back to work—and that is a big if—we could really run up the score with some big wins. But if he keeps picking and picking, it will be that much harder.”

And despite Trump’s troubles, GOP members still cannot put too much distance between him and them without risking voter backlash. “If anything, Trump’s diehard base is more sensitive about any slight of their messiah,” said the aide.