The Higgs mechanism

 What is called the Higgs mechanism is the extension of the spontaneous symmetry breaking to create massive vector bosons in a gauge invariant theory. Here it will be shown for a U(1) theory. The idea is to replace the normal derivative in (2.10) with the covariant derivative

 

$$\mu \partial_{\mu}^{} \mu$$ (14)

Adding the Lagrangian of the free gauge field A_$\mu$ , results in

 

$$\mu \varphi^{\ast}_{} \mu \varphi \varphi {\textstyle\frac{1}{4}} \mu \nu \mu \nu$$ (15)

This new Lagrangian is now invariant under the U(1) gauge transformation
 $$\begin{align} \varphi(x) \rightarrow\varphi'(x) &= \varphi(x) e^{{i}q \xi(x)} \... ...\\ A_\mu(x) \rightarrow A_\mu'(x) &= A_\mu(x) + \partial_\mu \xi(x),\end{align}$$
with $\xi$ any differentiable function. Continuing in exactly the same way as for the Goldstone model with a negative $\mu^{2}_{}$ and expressing the Lagrangian in terms of the variables $\sigma$ and $\eta$ as defined in (2.15) the result is
 $$\begin{align} {L}&= \frac{1}{2}\partial^\mu\sigma \partial_\mu\sigma -\lambda ... ...tial_\mu \eta \notag \\ & \quad + \text{higher order terms}. \notag\end{align}$$
The Lagrangian clearly has a massive vector boson field A and two scalar fields $\sigma$ , $\eta$ with $\eta$ massless, but unfortunately also a term A ^$\mu$$\partial_{\mu}^{}$$\eta$ which does not fit in. It can not be understood as a pertubative interaction term since it is quadratic in the fields as the terms for the free field are. However, a careful analysis [7] shows that the Lagrangian (2.22) has one degree of freedom too much. This extra degree of freedom can be absorbed by choosing a specific gauge, ie. performing a gauge transformation of the type (2.21), where $\varphi$(x) has the form

 

$$\varphi {\frac{\sqrt{2}}{2}} \sigma$$ (16)

Such a gauge transformation is always possible and the chosen gauge is called the unitary gauge. In this gauge the $\eta$ field disappears and what is left is the Lagrangian
 $$\begin{align} {L}&= \frac{1}{2}\partial^\mu\sigma \partial_\mu\sigma -\lambda ... ...2}q^2v^2 A_\mu A^\mu \\ & \quad + \text{higher order terms}. \notag\end{align}$$
In summary, it is seen that a complex scalar field and a massless vector field, both with two degrees of freedom, in (2.20) as a result of the Higgs mechanism was transformed in (2.24) into one real scalar field with one degree of freedom and a massive vector boson field with 3 degrees of freedom. A massless spin 1 particle has two transverse polarised states while a massive spin 1 particle has an additional longitudinal polarised state. It should be noted that the $\eta$ field only disappears if the $\eta$ bosons are massless. As shown in section 2.1.1 this requires the vacuum state to be degenerate ie. the Higgs mechanism will only work with a degenerate vacuum.

The Higgs mechanism was demonstrated here for a U(1) gauge invariant Lagrangian. To extend it to the SU(2) x U(1) gauge invariant Lagrangian of the electroweak theory is relatively simple. The starting point is a Lagrangian with a complex scalar doublet and four massless vector bosons. Counting degrees of freedom gives four from the scalars and eight from the vector bosons.

Through the Higgs mechanism the Lagrangian is transformed into one real scalar, three massive vector and one massless vector boson. The massless vector boson is of course to be identified with the photon and the single remaining scalar with the Higgs boson. Counting degrees of freedom again gives one from the Higgs, two from the photon and nine from the massive vector bosons, again adding up to twelve.

Introducing the masses of the vector bosons with one doublet of complex scalars is the simplest scenario, in principle an infinite number of scalar fields can be introduced. The simplest supersymmetric models instead have five scalar fields left after the Higgs mechanism: a doublet of charged scalars, two neutral scalars and one neutral pseudoscalar.

The masses of the particles in the standard model are given as

 

$$\sqrt{2\lambda} {\textstyle\frac{1}{2}} {\frac{m_{W}}{\cos\theta_{W}}}$$ (17)

where g is the weak coupling constant and $\theta_{W}^{}$ the Weinberg angle. Using

 

$${\frac{\sqrt{2}}{2G_{f}}} \alpha {\frac{g^2 \sin^2\theta_{W}}{4\pi}}$$ (18)

where G_F is the Fermi constant and $\alpha$ the fine structure constant, the vector boson masses can be expressed through G_F , $\alpha$ and sin $\theta_{W}^{}$ . With the Fermi constant measured from the muon lifetime and the Weinberg angle from the relative cross sections of neutral current ( $\nu_{\mu}^{}$ + p $\rightarrow$ $\nu_{\mu}^{}$ + X ) and charge current ( $\nu_{\mu}^{}$ + p $\rightarrow$ $\mu$ + X ) processes it was possible to predict the masses of the vector bosons. Their discovery at the UA1 and UA2 experiments at the CERN Sp $\bar{p}$S was a great victory for the electroweak theory.

The vacuum expectation value is easily extracted as

 

$${\frac{2m_{W}}{g}}$$ (19)

but there is no way to measure the value of $\lambda$ before a discovery of the Higgs. As discussed in section 2.5 self consistency of the standard model sets an upper limit on the Higgs mass around 1 TeV. Experimental Higgs mass limits are discussed in section 2.4.

For reference the full electroweak and Higgs sector part of the standard model Lagrangian is given. It takes the form

 

L = L ₀ + L _FB + L _FH + L _BB + L _BH + L _HH (20)

where L ₀ is the Lagrangian of the free fields
 $$\begin{align} {L}_0 &= \bar{\psi}_{f}({i}\partial\!\!\!/ - m_{f})\psi_{f} \no... ...l^\mu\sigma \partial_\mu\sigma - \frac{1}{2}m_{H}^2 \sigma^2, \notag\end{align}$$
and the remaining terms the interaction between fermions and bosons, fermions and the Higgs, bosons and bosons, bosons and the Higgs and finally Higgs self interactions. Only the fermion Higgs interaction term

 

$${\textstyle\frac{1}{v}} \bar{\psi}_{f}^{} \psi_{f}^{} \sigma$$ (21)

which will be important later is given here. A sum over all fermions is implicitly assumed in (2.29) and (2.30) with some terms obviously disappearing if the neutrinos are massless.

A remarkable feature of introducing massive fermions in the theory is that their interaction terms in the Lagrangian become proportional to their mass as seen in (2.30). While this makes it impossible to put constraints on the Higgs mass and the fermion masses, it has the consequence that the Higgs particle will couple to the fermions in proportion to their mass. The supersymmetric extensions of the Standard Model have different predictions for the coupling constants and hence a measurement of several coupling constants can be used to look into the theory behind the Higgs mechanism.