# So, what about the Standard Model?

It is now possible to state that the Standard Model is a gauge theory invariant under $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y$ symmetry group given by the direct product of the symmetry group of the strong interaction, $SU(3)_C$, with that of electroweak interaction, $SU(2)_L \otimes U(1)_Y$ which also contains the symmetry group of electromagnetic interaction $U(1)_{EM}$.

The general form involving these fields is

where:

• $Q,\ L,\ E,\ U,\ D$ indicate leptons and hadrons for each family order and the index $m$ runs over the three families of fermions;
• $G^\alpha_\mu$ with $\alpha=1,\dots,8$ represents the gluon fields coming from the invariance under $SU(3)_C$;
• $W^\alpha_\mu$ with $\alpha=0,1,2$ are the fields coming from the invariance under $SU(2)_L$;
• $B_\mu$ is the field coming from the invariance under $U(1)_Y$.

The quantum numbers related to these transformations are colour, isospin and hypercharge respectively. The covariant derivative is:

$\mathcal{D}_\mu=\partial_\mu-ig_1YB_\mu-ig_2\frac{\tau^\alpha}{2}W^\alpha_\mu-ig_3\frac{\lambda^b}{2}G^b_\mu,$

where $g_{1,2,3}$ are the coupling strength in the time–space of the Electromagnetic, Weak and Strong Interactions and $Y, \tau, \lambda$ are the corresponding operators.

The associated mediators to the symmetry group are 12 spin–1 bosons: 8 gluons, 3 $W_i$ bosons and 1 $B$ boson. The $W^\pm$ bosons are obtained as a linear combination of the $W^1_\mu$ and $W^2_\mu$ fields, while the $Z^0\$ boson and the photon $\gamma$ are obtainable as a linear combination of $W^0_\mu$ and $B_\mu$.
Gluons, the mediators of the strong interaction, are massless and neutral. They carry only the colour charge, so they can interact with each other; the $W^\pm$ and $Z^0$ bosons are massive particles, while the photon is massless.

Constituents of matter (i.e. fermions) can be distinguished into quarks and leptons.
Even the leptons are six: $e$, $\mu$, $\tau$ carry electric charge and present both electromagnetic and weak interactions, while $\nu_e$, $\nu_\mu$, $\nu_\tau$ are neutral and they are sensitive only to weak interactions.
To each fermion there is an associated anti–fermion with the same mass, spin and quantum numbers but opposite electric charge.
Anti–quarks has yet opposite colour charge with respect to the corrispondent quark.
Quantum numbers are also defined for fermions, baryons and leptons. The quarks have $L=0$ leptonic quantum number, but they carry $B=\frac{1}{3}$ baryonic quantum number, while anti–quarks have $B=-\frac{1}{3}$.

As a consequence, baryons have $B=1$ while mesons have $B=0$. On the contrary, leptons have $L=1$, anti–leptons have $L=-1$ and both they have $B=0$.
In addition, lepton family numbers $L_e$, $L_\mu$ and $L_\tau$ are also defined with the same assigning scheme: $+1$ for particles of the corresponding family, $-1$ for the anti–particles and $0$ for leptons of other families or non–leptonic particles.

Then, within the SM there are 61 fundamental particles: 24 fermions and the corresponding 24 antifermions, 12 vector bosons and just one scalar boson, the Higgs boson. Leptons and quarks can be grouped into generations each containing two quarks, one with electric charge $+\frac{2}{3}|e|$ and the other with electric charge $-\frac{1}{3}|e|$, and two leptons, one charged and the other neutral.

The generations observed so far are three differing only in mass that progressively grows from the first to the third generation.
Ordinary matter is composed only by the first generation particles: $u$, $d$, $e$, $\nu_e$.