In the Standard Model, the W and Z bosons and fermions get their masses via the Brout-Englert-Higgs mechanism. To this end, the Lagrangian contains a scalar field

*H*with a negative mass squared

*V = - m^2 |H|^2*. We know that the value of the parameter

*m*is around 90 GeV - the Higgs boson mass divided by the square root of 2. In quantum field theory, the mass of a scalar particle is expected to be near the cut-off scale

*M*of the theory, unless there's a symmetry protecting it from quantum corrections. On the other hand,

*m*much smaller than

*M,*

Relaxation is a genuinely new solution, even if somewhat contrived. It is based on the following ingredients:

- The Higgs mass term in the potential is
*V = M^2 |H|^2*. That is to say, the magnitude of the mass term is close to the cut-off of the theory, as suggested by the naturalness arguments. - The Higgs field is coupled to a new scalar field - the relaxion - whose vacuum expectation value is time-dependent in the early universe, effectively changing the Higgs mass squared during its evolution.
- When the mass squared turns negative and electroweak symmetry is broken, a back-reaction mechanism should prevent further time evolution of the relaxion, so that the Higgs mass terms is frozen at a seemingly unnatural value.

Then the story goes as follows. The axion Φ starts at a large value such that the Higgs mass term is positive and there's no electroweak symmetry breaking. During inflation its value slowly decreases. Once

*gΦ < M^2*, electroweak symmetry breaking is triggered and the Higgs field acquires a vacuum expectation value. The crucial point is that the height of the axion potential

*Λ*depends on the light quark masses which in turn depend on the Higgs expectation value

*v*. As the relaxion evolves,

*v*increases, and

*Λ*also increases proportionally, which provides the desired back-reaction. At some point, the slope of the axion potential is neutralized by the rising Λ, and the Higgs expectation value freezes in. The question is now quantitative: is it possible to arrange the freeze-in to happen at the value

*v*well below the cut-off scale

*M?*

*It turns out the answer is yes, at the cost of choosing strange (though not technically unnatural) theory parameters. In particular, the dimensionful coupling g between the relaxion and the Higgs has to be less than 10^-20 GeV (for a cut-off scale larger than 10 TeV), the inflation has to last for at least 10^40 e-folds, and the Hubble scale during inflation has to be smaller than the QCD scale.*

*θ*-term in the Standard Model Lagrangian. But here it is stabilized at a value determined by its coupling to the Higgs field. Therefore, in the toy-model, the axion effectively generates an order one

*θ*-term, in conflict with the experimental bound

*θ*< 10^-10. Nevertheless, the same mechanism can be implemented in a realistic model. One possibility is to add new QCD-like interactions with its own axion playing the relaxion role. In addition, one needs new "quarks" charged under the new strong interactions. These masses have to be sensitive to the electroweak scale

*v*, thus providing a back-reaction on the axion potential that terminates its evolution. In such a model, the quantitative details would be a bit different than in the QCD axion toy-model. However, the "strangeness" of the parameters persists in any model constructed so far. Especially, the very low scale of inflation required by the relaxation mechanism is worrisome. Could it be that the naturalness problem is just swept into the realm of poorly understood physics of inflation? The ultimate verdict thus depends on whether a complete and healthy model incorporating both relaxation and inflation can be constructed.

Certainly TBC.

Thanks to Brian for a great tutorial.

## 14 comments:

Hi, just to state the obvious: The correct link to the article is http://arxiv.org/abs/1504.07551

(Feel free to remove this comment after fixing.)

Oops, thanks, corrected.

In your introduction it seems to be a logic hole. Why one needs to "Protect" the parameter m from quantum corrections (at >-100-GeV scale)?

In quantum field theory scalar masses are expected to be near the cut-off scale of the theory, m~M, unless there's a symmetry that protects them (for example, pion in low-energy QCD). That's why naturalness typically implies that the SM should be replaced by another theory at 100 GeV. In this new theory m should be protected, otherwise it would have the same naturalness problem as the SM.

Speaking of logic holes in the context. [And I was greatly encouraged by the comment instructions of "insulting the author of this blog is allowed and even encouraged", guess it means a fair trade of insults is to be expected?]

But for a layman that comes to Jester's court for clearly laid out facts, the claim that "[anthropics is] to seek solace in religion" seems an untenable juggling of claims. [Say, by being mentioned here or in the first sentence in the Graham et al paper, instead of in a theological journal.] Much as untenable as Dine's plaint that since anthropics is difficult that is somehow a fault instead of an interesting feature. [Or "impossible", but I think Weinberg's anthropic CC paper rejects that, especially if that model later failed in the face of better observations as far as I know.]

To patch these holes, and make the logic crystal clear, wouldn't it be better to claim that "It is ugly. Burn it with fire!" (At which point the arguable usefulness of such logic can start.)

Supersymmetry on the other hand is so beautiful - it seems to this layman - it ought to be fact. Who didn't order that? =D

Adding new fields and/or symmetries beyond SM may be an obvious solution for the naturalness problem, yet it remains unconvincing at this point for (at least) a couple of reasons:

a) no shred of evidence so far,

b) general inability to account for ALL open puzzles associated with the SM.

There are solutions to the naturalness problem that do not explicitly rely on BSM physics. Unfortunately, they are too much off-topic to be discussed here.

Antropic argument answers no scientific question. If the question is "why A (unnaturalness)"? then "from A (may be) follows B, B is true so" is no answer at all. Even if there's solid scientific argument that strong unnaturalness is required for emergence of intelligent observer (I doubt that), it'd still shed no light on the underlying physical cause for such unnaturalness.

Anthropics is not an ugly theory, but an excuse for the absence of a theory. The two statements, "God chose these constants" and "Multiverse chose these constants", are quantitatively equivalent (if not qualitatively). It is not an explanation, but a surrender of attempts to find one.

In this sense, "to seek solace in religion" is a very accurate description of what is going on. It's just that multiverse/anthropics crowd are living in a delusion that their stance is based on more than just blind faith in a "cosmological accident".

And to preempt some comments --- no, life could very well have existed in the same way as it does even if, say, neutrino masses were exactly zero. Anthropic arguments just cannot account for all the values of all constants.

Hi Jester,

Great post.

Regarding your line "The toy-model above ultimately fails: since the QCD axion is frozen at a non-zero value"...

I wouldn't call what fails a toy model.... I rather think that the minimal model with the idea is ruled out due to the smallness of strong CP phase and we have to build non-minimal and more contrived models.

Maybe what I am saying is just semantics....

That's right, this wasn't quite precise. I modified this part slightly.

Sorry, I'm confused by a couple things. At early times the last term you write is actually a coupling to free gluons via G^~G; the last term only assumes the form you write after the chiral condensate forms -- right? More importantly: at early times phi is large, not small, and the first term is positive *because* phi is big -- it becomes negative later because phi decreases (which happens simply because that's the direction of the energy minimum).

The rest of the discussion, about the kind of hidden features, is nice, though!

In this model inflation happens below the Lambda_QCD scale, so the cos potential is effectively present during inflation. As for the 2nd point: the discussion is phi-shift-invariant, but i agree it's more transparent to present it the way you suggest. I changed the text accordingly.

"10^40 e-folds" seems excessive. Do you mean 40 e-folds or ln(10^40) = 92.1... e-folds?

Unfortunately, the number is correct (give or take 10^10 that depends on assumptions about the cutoff scale). This is one of the problems with this idea.

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