2.5 Generations

Our description of the Standard Model is almost at an end. We have told you about its gauge group, ${G_{\mbox{\rm SM}}}$, its representation $F \oplus F^*$ on the the first-generation of fermions and antifermions, and a bit about how these fermions interact by exchanging gauge bosons, which live in the complexified adjoint rep of ${G_{\mbox{\rm SM}}}$. For the grand unified theories we are about to discuss, that is all we need. The stage is set.

Yet we would be derelict in our duty if we did not mention the second and third generation of fermions. The first evidence for these came in the 1930s, when a charged particle 207 times as heavy as the electron was found. At first researchers thought it was the particle predicted by Yukawa--the one that mediates the strong force between nucleons. But then it turned out the newly discovered particle was not affected by the strong force. This came as a complete surprise. As the physicist Rabi quipped at the time: ``Who ordered that?''

Dubbed the muon and denoted $\mu^-$, this new particle turned out to act like an overweight electron. Like the electron, it feels only the electromagnetic and weak force--and like the electron, it has its own neutrino! So, the neutrino we have been discussing so far is now called the electron neutrino, $\nu_e$, to distinguish it from the muon neutrino, $\nu_\mu$. Together, the muon and the muon neutrino comprise the second generation of leptons. The muon decays via the weak force into an electron, a muon neutrino, and an electron antineutrino:

$$\mu^- \to e^- + \nu_{\mu} + \overline{\nu}_e .$$

Much later, in the 1970s, physicists realized there was also a second generation of quarks: the charm quark, $c$, and the strange quark, $s$. This was evidence of another pattern in the Standard Model: there are as many flavors of quark as there are leptons. In Section 3.3, we will learn about the Pati-Salam model, which explains this pattern by unifying quarks and leptons.

Today, we know about three generations of fermions. Three of quarks:

Quarks by Generation
1st Generation		2nd Generation		3rd Generation
Name	Symbol	Name	Symbol	Name	Symbol
Up	$u$	Charm	$c$	Top	$t$
Down	$d$	Strange	$s$	Bottom	$b$

and three of leptons:

Leptons by Generation
1st Generation		2nd Generation		3rd Generation
Name	Symbol	Name	Symbol	Name	Symbol
Electron	$\nu_e$	Muon	$\nu_{\mu}$	Tau	$\nu_{\tau}$
neutrino		neutrino		neutrino
Electron	$e^-$	Muon	$\mu^-$	Tau	$\tau^-$

The second and third generations of quarks and charged leptons differ from the first by being more massive and able to decay into particles of the earlier generations. The various neutrinos do not decay, and for a long time it was thought they are massless, but now it is known that some and perhaps all of them are massive. This allows them to change back and forth from one type to another, a phenomenon called `neutrino oscillation'.

The Standard Model explains all of this by something called the Higgs mechanism. Apart from how they interact with the Higgs boson, the generations are identical. For instance, as representations of ${G_{\mbox{\rm SM}}}$, each generation spans another copy of $F$. Each generation of fermions has corresponding antifermions, spanning a copy of $F^*$.

All told, we thus have three copies of the Standard Model representation, $F \oplus F^*$. We will only need to discuss one generation, so we find it convenient to speak as if $F \oplus F^*$ contains particles of the first-generation. No one knows why the Standard Model is this redundant, with three sets of very similar particles. It remains a mystery.