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 small value where the

*M^2*term dominates and there's no electroweak symmetry breaking. During inflation its value slowly increases. 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.