48.3 Spontaneous symmetry breaking

48.3 Spontaneous symmetry breaking

The Higgs field is an R⁴-valued scalar field, with a complex structure chosen so that it can be taken to be ${\mathbf{C}}^2.$ -valued $\Phi (\mathbf{x})$ , with $U(1)$ and $SU(2)$ acting by the defining representations. The Hamiltonian is

$$h_{Higgs}={\int }_{{\mathbf{R}}^3}(|\Pi {|}^2+|(\nabla -i\mathbf{A})\Phi {|}^2-m^2|\Phi {|}^2+\lambda |\Phi {|}^4)d^3\mathbf{x}$$

where $A$ are vector potential fields for $U(1)\otimes SU(2)$ acting in the defining representation, scaled by the coeficients $g_1,g_2$

Note the unusual sign of the mass term and the existence of a quartic term. This means that the dynamics of such a theory cannot be analyzed by the methods we have used so far. Thinking of the mass term and quartic term as a potential energy for the field, for constant fields this will have a minimum at some non-zero values of the field. To analyze the physics, one shifts the field $\Phi$ by such a value, and approximates the theory by a quadratic expansion of the potential energy about that point. Such a theory will have states corresponding to a new scalar particle (the Higgs particle), but will also require a new analysis of the gauge symmetry, since gauge transformations act nontrivially on the space of minima of the potential energy. For how this “Anderson-Higgs mechanism” afects the physics, one should consult a standard textbook.

Finally, the Higgs field and the spinor fields are coupled by cubic terms called Yukawa terms, of a general form such as

$$h_{Yukawa}={\int }_{{\mathbf{R}}^3}M\Phi {\Psi }^{†}\Psi d^{3}x$$

where $\Psi ,{\Psi }^{†}$ are the spinor fields, $\Phi$ the Higgs field and M a complicated matrix. When one expresses this in terms of the shifted Higgs field, the constant term in the shift gives terms quadratic in the fermion fields. These determine the masses of the spin $\begin{array}{l}\frac{1}{2}\end{array}$ particles as well as the so-called mixing angles that appear in the coupling of these particles to the gauge fields.

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