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The existence of asymmetric solutions to a symmetric theory is common to many branches of physics. The reason lies in the fact that the symmetric state is not the state of minimum energy, i.e., the ground state, and that in the process of evolving towards the ground state, the intrinsic symmetry of the system has been broken. Figure 06e shows that the initial position of the marble on top of the bump is symmetric but not in a state of | |
Figure 06e Symmetry Breaking [view large image] |
minimum energy. A small perturbation will cause the rotational symmetry to be broken and the system to assume the ground state configuration. When the symmetry of a physical system is broken in this way, it is often referred to as "spontaneous symmetry breaking". |
This idea can be applied to account for the mass of gauge boson in the electroweak interaction in the Standard model. Let us start by examine how such unstable symmetry can arise mathematically. Considering the Lagrangian for a scalar field with the potential V: L = (1/2)_{} - V(_{}) ---------- (46) where V(_{}) = (1/2)m^{2}_{}^{2} + (/4!)_{}^{4}, and 4! = 4x3x2x1. | |
Figure 06f Scalar Field |
If m^{2} > 0, the system has real mass, the potential exhibits a minimum at the origin, where _{} = 0. This system is associated with a unique vacuum _{} (see Figure 06f). |
The formulation can be generalized to complex scalar field with two independent components corresponding to positively and negatively charged fields. In a slightly different notations, the Lagrangian for a complex scalar field has the form similar to Eq.(46): L = (_{}_{}_{})(_{}^{}_{}*) - V(_{},_{}*) ---------- (48) where V(_{}) = m^{2}_{}_{}* + (/4)(_{}_{}*)^{2}. | |
Figure 06g Complex Scalar Field [view large image] |
As each term in the Lagrangian of the Standard model represents a different process, Figure 06h shows the various Higgs interactions in the form of Feynman diagrams. Diagram (a) represents a fermion emitting or absorbing a Higgs particle. Diagram (b) shows the corresponding process for the gauge bosons. They can also interact simultaneously with two Higgs, as shown in (c), which also represents a gauge boson scattering a Higgs particle. The Higgs also interacts with itself, as shown in diagrams (d) and (e), which are related to the shape of the scalar potential (Figure 06g). Diagram (f) depicts an electron acquiring its mass. | |
Figure 06h Higgs Field Interaction [view large image] |
By taking into account the higher term of the SM (Standard Model) perturbation series, the vacuum potential of the Higgs field V is in the form :
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Figure 06i Stability of the Universe [view large image] |
and the sum is over all SM particles acquiring a Higgs-dependent mass M_{i}. The precise form of V_{1} is not important in the present context, it just shows that the Higgs potential also depends on the particles it acts upon. Furthermore, only the heaviest top quark in the sum is retained in the following consideration. |