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<
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.
These 3 ingredients can be realized in a toy model where the Standard Model is coupled to the QCD axion. The crucial interactions are
The toy-model above ultimately fails: since the QCD axion is frozen at a non-zero value, one effectively generates an order one CP violating θ-term in the Standard Model Lagrangian, 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. TBC, for sure.
Thanks to Brian for a great tutorial.