### Recall from last lecture:

Quantum Electrodynamics (QED): the basic interaction vertex

Quantum Chromodynamics (QCD): the basic interaction vertex

The Weak Force: the basic interaction vertices

The fundamental charged vertex couples a W with an up-type quark and a down-type quark.
The up-type and down-type quarks can come from *different* generations, *i.e.* u with s, d with c, or c with b.
The allowed amount of cross-generational coupling is determined by the Cabibo-Kobayashi-Maskawa (CKM) matrix.

The picture is that the quarks that form hadrons are mass eigenstates, while what couples to the W and Z are the weak eigenstates, and for the quarks the two are not lined up.
By convention we choose our phase such that the up-type quarks have no mixing between mass eigenstates and weak eigenstates.
Then the weak eigenstates of the down-type quarks are linear combinations of the mass eigenstates given by:

d' | = | V_{ud} | V_{us} | V_{ub} | d |

s' | V_{cd} | V_{cs} | V_{cb} | s |

b' | V_{td} | V_{ts} | V_{tb} | b |

With the U/D notation defined above, we can write this as
D'_{i} = S_{j}V_{ij}D_{j}

The nine elements of the CKM matrix are complex numbers -- therefore 18 separate real numbers are needed to specify the matrix.
However, the matrix is unitary, meaning that
**VV**^{*} = **V**^{*}**V** = **1**

This relation yields 12 independent equations among the 18 numbers, and additionally we are free to choose two arbitrary phases.
We are left with just 4 numbers that describe this matrix.
### Electroweak

Some additional diagrams are needed to complete the electroweak theory.
The two most important are the WW

Copyright © Robert Harr 2003