How many generations and fermions are there in the Standard Model Group?

[shiraz]

Dear All
The standard Model Group is SU(3)*SU(2)*U(1) i know that there is 3 families and each family contain 16 fermion. I am trying to guess the number of families and fermions from the representation of the concerned group. For example the SO(10) How many generations we have and how may fermions? I will be so thankful if anyone could help me to find a criteria to study that in a simple way

 

[king vitamin]

The gauge groups are totally independent of the number of particles coupled to them. You can decide to couple matter to any representation of the gauge group, and that will change the number of fermions/bosons accordingly.

(There is a demand that there isn't a "gauge anomaly," which rears its head in the Standard Model because of the coupling to chiral fermions. But the Standard Model is not the unique way to cancel a gauge anomaly.)

 

[Orodruin]

and each family contain 16 fermion

Just to point out that the SM contains 15 fermions per family:

• 2x3=6 left-handed quarks
• 3 right-handed up type quarks
• 3 right-handed down type quarks
• 2 left-handed leptons
• 1 right-handed charged lepton

6+3+3+2+1 = 5x3 = 15
However, many extensions of the SM also include a right-handed neutrino, making a total of 16, but the RH neutrino is a priori not a part of the SM. These fermions (including the RH neutrino) fit well into the 16 representation of SO(10).

What does change with the number of generations are the beta functions and therefore the running of the coupling constants. For example, if you slam too many (17) fermion generations into QCD, the beta function will no longer be negative (it contains a factor $11 - 2n_f/3$).