Physics 357 : 1998 Lecture 21

Generations and Weak Isospin

The weak interaction couples the fermions in pairs, and the pairs are usually described as belonging to different generations

The relative assignment of the quark and lepton generations is just according to their mass.

This two-state pairing is reminisent
of the the two spin states allowed for a spin 1/2 fermion, or
the 2 isospin states of a nucleon (proton and neutron), so we
define all known fermions to have "weak isopin" 1/2,
and the "up" type fermions are assigned T_{3}=+1/2,
and the "down" type fermions are assigned T_{3}=-1/2.

Electroweak Theory

The group describing the spin states of a fermion is SU(2). So an SU(2) gauge theory is the "obvious" theory for the weak interaction. A gauge theory needs bosons to transform between all types of "charge" in the theory, i.e.

W^{+}: T_{3}=+1
{(T_{3}=^{1}/_{2}) ->
(T_{3}=-^{1}/_{2})},
e.g.

W^{-}: T_{3}=-1 {(T_{3}=-^{1}/_{2})
-> (T_{3}=^{1}/_{2})},
e.g.

W^{0}: T_{3}=0 ·__(__+^{1}/_{2}-^{1}/_{2}__)__-__
(__-^{1}/_{2}+^{1}/_{2}__)__Ò
{(T_{3}=±^{1}/_{2}) ->
(T_{3}=±^{1}/_{2})},
e.g.

But if this is true, then why is the rate for

9 orders of magnitude smaller than

Glashow, Iliopoulos, and Maiani showed (in 1970, before charm was discovered) that Cabibbo mixing along with the existence a charmed quark solves this problem. (See F&H equation 13.10.)

The W^{0} has the same quantum
numbers as the photon, so mixing occurs leading to a unified
electroweak theory.

Pure SU(2)_{L} Bosons (Weak
Isospin=1)

Pure U(1) Boson (Weak Isospin=0)

The actual observed bosons are the
g
and the Z^{0}.

Both the Z^{0} and the g
are part SU(2) and part U(1) gauge bosons, so the coupling of
the Z^{0} depends on both the electric charge and the
T_{3} of the fermions. i.e. The coupling of the Z^{0}
to "up" type quarks is different from the coupling to
"down" type quarks, which is different from charged
leptons, which is different from neutrinos.

Both the Z^{0} and the g
couple only fermions to themselves.

e.g. Only

Never

Predictions of electroweak theory

**Neutral currents exist**

e.g. (F&H equations 13.42, 13.43)

Neutral currents discovered 1973 at CERN.

**W**, with^{±}and Z^{0}bosons exist

(F&H equations 13.37, 13.38)

The W & Z were discovered in 1982 at CERN.

**the top quark must exist**

or else the b quark will not have full weak coupling

The top quark was discovered in 1994 by the CDF experiment at Fermilab.

Some questions about electroweak gauge theory:

- Why is it left handed? SU(2)
_{L}!

- Why is gauge symmetry broken? M
_{W},M_{Z}>0! Masses? - How is unitarity violation resolved? s(WW->WW) ~ s!

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