## 1 Introduction

The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a central pillar of the Electroweak Standard Model. Its origins are with the pioneering 1963 work of Cabibbo [1], who reconciled the concept of vector-current universality with the recently observed difference between strangeness changing and strangeness non-changing weak decays by proposing that universality was manifest only if one considered the total strength of both the strangeness changing and non-changing decays. In modern terminology we would say that he was the first to realize that there was mixing between the first two generation of quarks. Ten years later, Kobayashi and Maskawa [2], who were faced with the puzzle of CP violation, demonstrated that it could only be explained by the existence of a third generation of quarks with many more mixing possibilities. They argued that the quark-mass eigenstates are not the same as the weak-interaction eigenstates, but are related by a 3× 3 unitary rotation matrix – the CKM matrix.

By convention, the three quarks with charge 2/3 (*u*, *c* and *t*) are unmixed and all the mixing is expressed in terms of the CKM matrix operating on the quarks with charge -1/3 (*d*, *s* and *b*):

The Standard Model itself does not prescribe the elements of the CKM matrix – they have to be determined experimentally – but the Standard Model does insist that the matrix be unitary. This requirement can be tested if the elements of the matrix can be individually measured with sufficient precision. Should the result violate the matrix's unitarity, then that would be very exciting. It would indicate that the Standard Model is flawed, or approximate, and that some “new physics” must be injected. To date there is no evidence for this. Nevertheless unitarity tests continue to be pursued with ever-increasing precision since they provide limits on any possible new physics, and these limits themselves yield interesting conclusions.

There are two types of tests for CKM unitarity:

- Test the orthogonality of rows and columns: (2)It is difficult to achieve high precision with this type of test since six elements are involved, from all three quark generations. Though it is currently a very active field of research, the results cannot yet match the precision achieved by normalization tests. We will not pursue such tests any further here.
- Test the normalization of rows and columns: (3)For the first row or first column, this normalization test is made much easier by the realization that the contribution from the third-generation member is negligibly small, so only two matrix elements must be determined precisely. To date, the best precision is obtained in the top-row test(4)

In this article, we will concentrate on the top-row normalization test beginning with a discussion of the determination of from nuclear beta decay.