Lepton– symmetry and mixing Now we will discuss the with participation of . As we mentioned in the last lecture, the weak interactions of quarks may be of two types: purely hadronic and semileptonic. The purely hadronic weak decays are not nearly so well understood as the semileptonic ones, because the final-state interact strongly with each other, and effects are very difficult to calculate. We will concentrate here mainly on the semileptonic interactions (but some material is applicable to purely hadronic weak interactions as well, and we will mention it) We will initially restrict ourselves to the weak interaction involving first two generations of quarks:

In the next lecture we will cover the third generation weak interactions. Lepton–quark symmetry and mixing (2) Two ideas are the most Important in weak interactions of quarks: 1) lepton–quark symmetry and 2) quark mixing. We will start from the lepton quark symmetry. In simplest form (for two generations) it states that the two generations of quarks − − and the related two generations of leptons, νe, e and νμ , μ , have identical weak interactions. In this case one obtains the basic W– quark vertices by the − − replacements νe → u, e → d, νμ → c, and μ → s in the basic W– lepton vertices leaving the weak coupling constant gW unchanged. The result is on Fig 9.10 on the right. Fig 9.10, Quark vertices The quark lepton symmetry (without quark mixing) works from reasonably well numerically e.g. for prediction of the e.g. quark-lepton decay, but does not predict numerically without the introducing symmetry, quark mixing the decays (see diagrams on the next slide) with quark mixing ignored Lepton–quark symmetry and mixing (3)

The second idea (of quark mixing) is that d and s quarks participate in the weak interactions via the linear combinations below, where the parameter θC is called the Cabibbo angle, this angle is purely phenomenological (taken from experiment):

In other words, lepton quark symmetry has to be applied to the modified doublets: Lepton–quark symmetry and mixing (4)

Fig. 9.12. Quark mixing, some diagrams The u and d quark modified charged weak interactions, with the Cabibbo quark mixing included. In Fig. 9.13 (below) the resulting from mixing additional diagrams with s->u and c->d transitions are shown

Experimentally determined value of Cabibbo angle is Since the value of Cabibbo angle is small, amplitudes of processes, proportional to sine of Cabibbo angle, are called Cabibbo suppressed, while amplitudes proportional to cosine of the Cabibbo angle are called Cabibbo allowed (in the probabilities it is squared as usual) : Lepton–quark symmetry and mixing (5)

Now we will look at the charmed quark couplings gcd and gcs. They are measured most accurately in scattering and yield a value consistent with the Cabibbo angle. The most striking result here is for charmed decays, which almost always produce a in the final state. Again the related terminology is that since the value of Cabibbo angle is small, some processes are Cabibbo allowed while others are supressed. The decay of a charmed will almost always lead to a strange hadron in the final state ( see e.g. diagrams on the next slide). Examples of the dominant Cabibbo allowed charm decays include (see Fig 9.14 on the right): W decays W can decay to leptons, and also from lepton–quark symmetry it can decay to quark–antiquark pairs (Cabibbo mixed as below), with the same coupling strength αW

Since quarks will be produced in 3 color sates, the ratio to one lepton pair:

There are 3 lepton pairs and universality between generations, so we conclude that branching ratios of hadronic and leptonic decays: Selection rules in weak decays Many observations of weak decays of are explained without the need for detailed calculations. As an example, the decays seem very similar, where Σ+(1189) = uus and Σ−(1197) = dds arethe charged Σ discussed. However, while reaction (9.27) is observed, reaction (9.28) is not. The experimental upper limit is

The reason is that the second process of Σ− decay is possible only in the second order in the weak interaction (no diagram with one W can be drawn), and it is negligibly small. ‘This example is just one of many processes which are allowed from the conservation laws but cannot proceed in the first order of weak interactions, and for this reason practically forbidden. Selection rules in weak decays (2) Several phenomenological selection rules involving possible changes in the combination of strangeness and charge In hadronic and semileptonic weak decays have been developed hystorically. They have been used very successfully well before the development of quarks and , are still valid today and can be used for testing of models. For detailed discussion of all the Selection rules see M&S book. E.g. a selection rule (9.32) which holds for all weak decays, was used for the decays of the so-called cascade particles, for Ξ0(1532) = ssu, Ξ−(1535) = ssd. (S = −2) and for the omega-minus decays Ω−(1672) = sss (S = −3),

Example of use of this rule in the very important prediction of the cascade decays of the omega-minus baryon discussed on the next slides (see M&S) Selection rules in weak decays (3) Selection rules in weak decays (4)