We know they have to be these matrices, up to normalization, because these act the right way on nucleons in C2:

+ p ! n

+ + n ! p0 + p ! p0 + n ! n:

Now, apply the standard isomorphism End(C2) C2 C2 to write these matrices

=

as linear combinations of quarks and antiquarks:

+ = u d; 0 = u u d d; = d u:

Note these all have the right I3, because isospins reverse for antiparticles. For example, I3(d) = + 12 , so I3( +) = 1.

In writing these pions as quarks and antiquarks, we have once again neglected to write the color, because this works the same way for all pions. As far as color

goes, pions live in

C3 C3 :

Con nement says that pions need to be white, just like nucleons, and there is only a one-dimensional subspace of C3 C3 on which SU(3) acts trivially, spanned by

r r + g g + b b 2 C3 C3 :

So, this must be the color state of all pions.

Finally, the Gell-Mann{Nishijima formula also still works for quarks, provided we de ne the hypercharge for both quarks to be Y = 13 :

Since nucleons are made of three quarks, their total hypercharge is Y = 1, just as before.

2.2.2Leptons

With the quarks and electron, we have met all the fundamental fermions required to make atoms, and almost all of the particles we need to discuss the Standard Model. Only one player remains to be introduced: the neutrino, . This particle completes the rst generation of fundamental fermions:

The First Generation of Fermions | Charge

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Neutrinos are particles which show up in certain interactions, like the decay of a neutron into a proton, an electron, and an antineutrino

n ! p + e + :

Indeed, neutrinos have antiparticles , just like quarks and all other particles. The electron's antiparticle, denoted e+, was the rst discovered, so it wound up subject to an inconsistent naming convention: the `antielectron' is called a positron.

Neutrinos carry no charge and no color. They interact very weakly with other particles, so weakly that they were not observed until the 1950s, over 20 years after they were hypothesized by Pauli. Collectively, neutrinos and electrons, the fundamental fermions that do not feel the strong force, are called leptons.

In fact, the neutrino only interacts via the weak force. Like the electromagnetic force and the strong force, the weak force is a fundamental force, hypothesized to explain the decay of the neutron, and eventually required to explain other phenomena.

The weak force cares about the `handedness' of particles. It seems that every particle that we have discussed comes in leftand right-handed varieties, which (quite roughly speaking) spin in opposite ways. There are are left-handed leptons,

which we denote as

L eL

and left-handed quarks, which we denote as

uL dL

and similarly for right-handed fermions, which we will denote with a subscript R. As the terminology suggests, looking in a mirror interchanges left and right|in a mirror, the left-handed electron eL looks like a right-handed electron, eR, and vice versa. More precisely, applying any of the re ections in the Poincare group to the (in nite-dimensional) representation we use to describe these fermions interchanges left and right.

Remarkably, the weak force interacts only with left-handed particles and righthanded antiparticles. For example, when the neutron decays, we always have

nL ! pL + eL + R

and never

nR ! pR + eR + L:

This fact about the weak force, rst noticed in the 1950s, left a deep impression on physicists. No other physical law is asymmetric in left and right. That is, no other physics, classical or quantum, looks di erent when viewed in a mirror. Why the weak force, and only the weak force, exhibits this behavior is a mystery.

Since neutrinos only feel the weak force, and the weak force only involves lefthanded particles, the right-handed neutrino R has never been observed directly. For a long time, physicists believed this particle did not even exist, but recent observations of neutrino oscillations suggest otherwise. In this paper, we will assume there are right-handed neutrinos, but the reader should be aware that this is still open to some debate. In particular, even if they do exist, we know very little about them.

Note that isospin is not conserved in weak interactions. After all, we saw in the last section that I3 is all about counting the number of u quarks over the number of d quarks. In a weak process such as neutron decay

udd ! uud + e + ;

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