# Spontaneous symmetry breaking

One of the most important missing aspects of the SM theory has been for a long
the origin of mass of fundamental particles.

The Lagrangian of the full Standard Model (*) well-describes the interactions of matter and radiation, but it does not include any mass term for the introduced bosons, while fermion masses are put by hand via an ad hoc Yukawa coupling.

The presence of mass terms would break the $SU(3)_C\otimes SU(2)_L\otimes U(1)_Y$ symmetry and would violate the gauge invariance of the theory.
According to the observations, while gluons are massless and the $SU(3)_C$ gauge invariance of strong interaction can be conserved, the $Z^0$ and $W^\pm$ bosons are massive.
The Lagrangian of the electroweak interaction cannot contain any mass term for both bosons and fermions.

For bosons, the addition of $M^2W_\mu W^\mu$ terms, which clearly depend from the choice of the gauge, explicity violates the gauge invariance and would lead to not renormalizable divergences in the theory; for fermions, mass terms would be of the form $-m\overline{\psi}\psi$ but such terms are not allowed in the Lagrangian as they are not gauge invariant and they contradict the left–handed structure of the weak interaction.
So the $SU(2)_L \otimes U(1)_Y$ symmetry, which describes the unified electroweak interaction has to be broken to allow to the $Z^0$ and $W^\pm$ bosons to have a mass. This is known as Spontaneous Symmetry Breaking (SSB).

The same mechanism also introduces the masses of the quarks and of the charged leptons.It is needed to ensure that terms of mass appear without explicitly violate the gauge symmetry, in order to keep the theory renormalizable and thus predictive. This is provided by the introduction of additional fields that spontaneously breaks the symmetry.

As it was proven, in this case the SM stays renormalizable.