# The Higgs mechanism

The simplest way to implement the SSB in the electroweak theory is achieved by adding the Higgs field, a $SU(2)$ complex scalar isospin doublet with four degrees of freedom,

Without affecting the gauge invariance, it is possible to add to the Lagrangian of the electroweak interaction (*) the term

$\mathcal{L}_{SSB}=(\mathcal{D}_\mu \Phi)^\dag (\mathcal{D}^\mu \Phi) -V(\Phi)$

which indicates the SSB, with $\lambda>0$ and $\mu^2<0$ in the potential $V(\Phi)$ explicited in the equation. The covariant derivative has the form

$\mathcal{D}^\mu=\partial^\mu-ig\frac{\sigma^i}{2}W_i^\mu-ig'YB^\mu$

where $\sigma^i$ indicates the Pauli matrices that are the generators for the group $SU(2)$ and $g$, $g'$ are the coupling constants associated to the fields $W$ and $B$.

The first term in the SSB contribute added to the Lagrangian is the usual kinetic energy term for a scalar particle for which the Euler-Lagrange equations lead to the Klein-Gordon equation of motion.
The potential $V(\Phi)$ that breaks the symmetry is known as the Higgs potential or mexican hat potential, $V(\Phi)=\mu^2 \Phi^\dag \Phi +\lambda(\Phi^\dag \Phi)^2$, due to its form.

The requirement $\mu^2 <0$ is necessary to ensure that the $\mathcal{L}_{SSB}$ admits an infinite number of degenerate minimum energy states not invariant under transformations of $SU(2)$.

In fact, the Higgs potential, showed in the picture, is symmetric and presents a local maximum in its centre. Thus, even the energy in the centre is symmetric but it constitutes an excited state that will still be unstable.
Because of the form of the potential, the stable final states at the minimum, the vacuum states,

$\langle\Phi^\dag\Phi\rangle_0=\frac{\mu^2}{\lambda}$

are not symmetric any more and this breaks the symmetry.
The requirements that the ground state must have a nonzero Vacuum Expectation Value (VEV) and it must be electrically neutral implies that $\Phi$ has to be of the form

with $v=\frac{\mu}{\sqrt\lambda}$.
Both the $SU(2)_L$ and the $U(1)_Y$ symmetry are broken, and the $U(1)_{EM}$ symmetries for the electric charge operator remains preserved.

At this point, the quantum fluctuations of the four degrees of freedom of the Higgs field $\Phi$ can be expanded around the ground state, as follows by a re–parameterization including an arbitrary $SU(2)$ phase factor

where the term $\zeta(x)$ indicates the (real) fields and corresponds to excitations of the field along the potential minimum. They coincide with the massless Goldstone bosons of a global symmetry, in this case $3 \times 3$ rotations of the $SU(2)$ group.
Nevertheless, in a local gauge theory, such massless bosons can be eliminated by a local $SU(2)$ rotation, corresponding to a unitary transformation named unitary–gauge (U-gauge):

Consequently, the fields $\zeta$ become of no physical significance. Three of the four degrees of freedom of the scalar doublet are absorbed by the gauge bosons and the remaining degree of freedom is associated with the scalar Higgs field. Only the real field $h(x)$ can be interpreted as a real particle, the Higgs boson.

The kinetic part of the SSB Lagrangian gives rise to quadratic terms in the wave function of the vector bosons. Thus the mass terms of the physical fields become

$M^2_W=\frac{1}{4}g^2v^2,\ \ \ M^2_Z=\frac{g'^2+g^2}{4}v^2$

where, as usual, $g$ and $g'$ are the gauge couplings of the $SU(2)$ and $U(1)$ groups of the weak and electromagnetic interactions, respectively.

The mass of the Higgs boson can be deduced from the SSB Lagrangian and it is given by

$m_H=\sqrt{-2\mu^2}=\sqrt{\frac{\lambda}{2}}v$

where $\lambda$ is the Higgs self–coupling parameter and $v$ is the VEV of the Higgs field. The Fermi coupling $G_F = \frac{\sqrt{2}}{2v}$ is determined with precision of $0.6\ ppm$ from muon decay measurements and it determines the value $v\approx 246\ GeV$.

Since $\lambda$ is unknown, the value of the SM Higgs boson mass $m_H$ cannot be predicted. Recent evidences at $\textsf{LHC}$ set its value at $125.9\pm 0.4\ GeV$.

At this point, the Lagrangian in the SM Lagrangian contains the mass terms for the gauge fields $W^\alpha_\mu$ and $B_\mu$.
To give rise to the mass of quarks and leptons, a further term is to be introduced in the Lagrangian to describe the coupling between the Higgs and the fermions. So, the Lagrangian of the Yukawa interaction is

$\mathcal{L}_{Yukawa}=-G_e(\overline{R}(\Psi^\dag L)+(\overline{L}\Psi)R)$

where $\Psi$ represents a generic fermion field obtained by the reparameterization showed in the relation for the U-gauge and by the application of the U-gauge in the above equation for the re-parameterization; $G_e$ stands for the coupling constant, while $L$ and $R$ and stand for the left and right functions respectively.

The presence of the field $\Psi$ between the fields $L$ and $R$ preserves the left–handed structure of the Lagrangian of the electroweak interaction. In this way, the $\mathcal{L}_{SSB}$ contains the coupling terms to the Higgs and the mass terms of the fermions that are quadratic in the fermion fields.

Through the SSB, these fermions acquire a mass proportional to the vacuum expectation value of the Higgs field. The heavier the particle, the stronger the coupling to the Higgs boson, according to the presence in the Yukawa’s theory of a single coupling parameter for all masses. Therefore, the Higgs “prefers” to decay into the heaviest kinematically allowed particles pair.