Weak decays

Quarks have an interesting property in that their weak eigenstates are mixtures of the mass eigenstates. By convention the down-type (, , and ) quarks are chosen to be mixtures of the mass eigenstates; these new weak eigenstates are denoted by $\dq^{\,\prime}$, $\sq^{\,\prime}$, and $\bq^{\,\prime}$. The up-type (, , and ) quarks are unmixed. This allows an up-type quark to decay, via the emission of a , to any energetically allowed down-type quarks. The reverse is also true.

For the simple two family case the transitions between quark types are described by the transformation postulated by Cabibbo [15]:

In this description the Cabibbo angle ($\theta_c$) has been measured to be approximately 0.23 radians. As a consequence the transitions $\cq \to \sq$and $\uq \to \dq$ are proportional to $\cos^2 \theta_c$ while the transitions $\cq \to \dq$ and $\sq \to \uq$ are proportional to $\sin^2 \theta_c$. The first set of transitions are called ``Cabibbo favored'' while the second set of transitions are denoted ``Cabibbo suppressed.'' (The value of $\sin^2 \theta_c / \cos^2 \theta_c$ is approximately $\frac{1}{20}$.)

With the discovery of a third family of quarks, the Cabibbo matrix was generalized and replaced with the CKM⁷ matrix [16] which gives the transition rates as

In this formalism, the transition rates between families are described by the values of V. For instance, the Cabibbo suppressed $\cq \to \dq$ transition rate is proportional to $\left\vert V_{cd} \right\vert^2$. In the CKM matrix, the diagonal elements are near unity, while the off diagonal elements are small. Assuming that there are only three families, the CKM matrix is a unitary matrix, which provides additional constraints on the values of V.

These contributions modulating quark transition probabilities arise from the coupling of the relevant quarks with with the boson. For instance, in the Cabibbo suppressed decay $\dzero \to \piplus \piminus$ decay shown in intro:wcoupling, the transition probability is proportional to $\left\vert V_{cd} \right\vert^2 \left\vert V_{ud} \right\vert^2$, one factor for each quark- vertex. Because of the large mass of the , the weak force has a very short range and consequently a very small magnitude. This effect is described by the Yukawa [17] potential which describes the range of a force mediated by a massive boson.⁸ Particles which decay via the weak nuclear force consequently have relatively long lifetimes.

  

Figure 1.1: Feynman diagram for a Cabibbo suppressed decay.