Chapter 14: Problem 1

Draw the lowest-order Feynman diagrams for the decays $$ K^{0} \rightarrow \pi^{+} \pi^{-}, \quad K^{0} \rightarrow \pi^{0} \pi^{0}, \quad \bar{K}^{0} \rightarrow \pi^{+} \pi^{-} \quad \text { and } \quad \bar{K}^{0} \rightarrow \pi^{0} \pi^{0} $$ and state how the corresponding matrix elements depend on the Cabibbo angle $\theta_{c}$.

Short Answer

Expert verified

The decays involve weak interactions changing quark content mediated by W bosons. The matrix elements depend on \theta_c\ via sin \theta_c\ and cos \theta_c\ factors.

Step by step solution

Understand the Decay Processes

The decay processes involve the weak interaction, as the strangeness is changing. Each of the states involves strange quarks turning into up quarks.

Identify the Quark Content

Identify the quark content of the particles involved: - $ K^{0} $ (sedbar) - $ \bar{K}^{0} $ (dsbar) - $ \pi^{+} $ (u dbar) - $ \pi^{-} $ (d ubar) - $ \pi^{0} $ (u ubar, d dbar)

Draw the Feynman Diagram for $ K^{0} \rightarrow \pi^{+} \pi^{-} $

Draw a diagram where a $ s $ quark in the $ K^0 $ decays via exchange of a $ W^+ $ boson into a $ u $ quark and an $ \bar{d} $. The $ W^+ $ decays into a $ d $ quark and an $ \bar{u} $. The final state particles are $ \pi^+ (u \bar{d}) $ and $ \pi^- (d \bar{u}) $.

Draw the Feynman Diagram for $ K^{0} \rightarrow \pi^{0} \pi^{0} $

Draw a similar diagram but considering that the final pion state has neutral particles which are superpositions of $ u \bar{u} $ and $ d \bar{d} $ pairs.

Draw the Feynman Diagram for $ \bar{K}^{0} \rightarrow \pi^{+} \pi^{-} $

Consider the initial state has a $ \bar{s} $ quark and a $ d $ quark. The $ \bar{s} $ quark decays via $ W^- $ boson into $ \bar{u} $ and the $ d $ quark. The $ W^- $ boson decays into $ u $ and $ \bar{d} $ which combines to form the final pion pairs.

Draw the Feynman Diagram for $ \bar{K}^{0} \rightarrow \pi^{0} \pi^{0} $

Draw a similar diagram as for $ K^{0} \rightarrow \pi^{0} \pi^{0} $ but with the initial state involving $ \bar{K}^0 $ quark content.

State Dependence on Cabibbo Angle

The matrix elements for these decays depend on the Cabibbo angle $ \theta_c $ through the factors associated with the quark decays. Specifically, the transitions $ s \rightarrow u $ and $ \bar{s} \rightarrow \bar{u} $ involve factors of $ \sin \theta_c $ and $ \cos \theta_c $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Weak Interaction

The weak interaction is one of the four fundamental forces in nature, responsible for processes that change the type (or 'flavor') of quarks. It's important in particle physics as it governs beta decay and other phenomena involving neutrinos and quarks. This interaction is mediated by the exchange of W and Z bosons. In kaon decays such as $ K^{0} \rightarrow \pi^{+} \pi^{-} $ and $ \bar{K}^{0} \rightarrow \pi^{0} \pi^{0} $, the weak force allows the transformation of a strange quark into an up quark. This happens through the exchange of a W boson, which is why Feynman diagrams for these decays involve W bosons transforming quarks.

Quark Content

Understanding the quark content of the particles involved in kaon decays is essential. Quarks combine in different ways to form protons, neutrons, and other particles known as hadrons. For instance:

$ K^{0} $ consists of a strange quark (s) and an anti-down quark (\bar{d}).
$ \bar{K}^{0} $ contains a down quark (d) and an anti-strange quark (\bar{s}).
$ \pi^{+} $ is made from an up quark (u) and an anti-down quark (\bar{d}).
$ \pi^{-} $ contains a down quark (d) and an anti-up quark (\bar{u}).
$ \pi^{0} $ is a mixture of up-antiup (u$ \bar{u} $) and down-antidown (d$ \bar{d} $) quark pairs.

By knowing these contents, we can accurately draw Feynman diagrams illustrating kaon decay processes.

Cabibbo Angle

The Cabibbo angle $ \theta_c $ is a parameter in the quark mixing matrix, which describes the probability of a quark transforming into another type via weak interaction. In the context of kaon decays, this angle affects the transition rates between quarks. For example:

$ s \rightarrow u $ transition involves the factor $ \sin \theta_c $.
$ d \rightarrow s $ transition involves the factor $ \cos \theta_c $.

When calculating the matrix elements of these decay processes, the Cabibbo angle plays a crucial role as it determines the strength of these quark transitions. Therefore, the probability amplitudes in the Feynman diagrams for kaon decays will depend on $ \sin \theta_c $ or $ \cos \theta_c $.

Particle Physics

Particle physics explores the fundamental constituents of matter and their interactions. It categorizes particles into quarks, leptons, gauge bosons, and more. Within this field, experiments and theories like Feynman diagrams help us understand particle interactions. Kaons are mesons (particles containing a quark and an antiquark) studied in particle physics to probe weak interactions and CP violation (differences in behavior between matter and antimatter).
This understanding helps develop our models of the universe's fundamental forces. Studying kaon decays gives insights into the Standard Model, especially the weak force, by observing how quarks change types through W bosons.

K Meson Decays

Kaons, or K mesons, are particles containing a strange quark paired with either an up or down antiquark. They can decay into pions ($ \pi $ mesons) via weak interaction. Key decays to consider are:

$ K^{0} \rightarrow \pi^{+} \pi^{-} $
$ K^{0} \rightarrow \pi^{0} \pi^{0} $
$ \bar{K}^{0} \rightarrow \pi^{+} \pi^{-} $
$ \bar{K}^{0} \rightarrow \pi^{0} \pi^{0} $

Feynman diagrams illustrate the transformation mechanism. For example, in $ K^{0} \rightarrow \pi^{+} \pi^{-} $, a strange quark turns into an up quark via a W boson, which then decays into a down quark and an anti-up quark. Understanding these processes requires knowledge of quark content, weak interactions, and the Cabibbo angle, all combining to explain how kaons transform into pions.

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Short Answer

Step by step solution

Understand the Decay Processes

Identify the Quark Content

Draw the Feynman Diagram for \( K^{0} \rightarrow \pi^{+} \pi^{-} \)

Draw the Feynman Diagram for \( K^{0} \rightarrow \pi^{0} \pi^{0} \)

Draw the Feynman Diagram for \( \bar{K}^{0} \rightarrow \pi^{+} \pi^{-} \)

Draw the Feynman Diagram for \( \bar{K}^{0} \rightarrow \pi^{0} \pi^{0} \)

State Dependence on Cabibbo Angle

Key Concepts

Weak Interaction

Quark Content

Cabibbo Angle

Particle Physics

K Meson Decays

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