1.2.2 Problems with the Standard Model

Once the $SU(3)_C \times SU(2)_L \times U(1)_Y$gauge structure and the mass generation mechanism are established this way, we may start seriously asking many unresolved questions within the Standard Model. Why do the electric charges of electron and proton exactly balance? Why are the strengths of the gauge interactions so different? Why is the number of generations three? Why do the seemingly independent anomalies from the quark sector and lepton sector cancel? Where do the fermion masses come from? Why is the CP invariance broken? And many others. Among them, the far most important question is: Why is the electroweak symmetry broken, and why at the scale $\langle H \rangle = 246$ GeV?

The Standard Model cannot answer any of these questions. This is exactly why we believe that there lies a more fundamental physics at a higher energy scale which leads to the unanswered characteristics of the Standard Model. Then all the parameters and quantum numbers in the Standard Model can be derived from the more fundamental description of Nature, leading to the Standard Model as an effective low-energy theory. In particular, the weak scale itself $\langle H \rangle = 246$ GeV should be a prediction of the deeper theory. The scale of the fundamental physics can be regarded as a cutoff to the Standard Model. Above this cutoff scale, the Standard Model ceases to be valid and the new physics takes over.

The mass term of the Higgs field is of the order of the weak scale, whereas the natural scale for the mass term is, however, the cutoff scale of the theory, since the quantum correction to the mass term is proportional to the cutoff scale squared because of the quadratic divergence. This problem, so-called the naturalness problem, is one of the main obstacles we encounter, when we wish to construct realistic models of the ``fundamental physics'' beyond the Standard Model. If the cutoff scale of the Standard Model is near the Planck scale, one needs to fine-tune the bare mass term of the Higgs potential to many orders of magnitude to keep the weak scale very tiny compared to the Planck scale. There are only two known possibilities to solve this problem. One is to assume that the cutoff scale of the Standard Model lies just above the weak scale and the other is to introduce a new symmetry to eliminate the quadratic divergence: supersymmetry. In the former scenario the Higgs boson mass tends to be heavy, if any, while in the latter it is expected to be light.