Draw possible Feynman diagrams for the following phenomena: a) protons scattering off each other b) neutron beta decays to a proton: \(n \rightarrow p+e^{-}+\bar{\nu}_{e}\).

To draw the Feynman diagram for protons scattering off each other, we need to consider their internal structure. A proton is composed of two up quarks and one down quark. The interaction between quarks in protons is mediated by the exchange of gluons, which are the carriers of the strong force. In this case, we can consider a simple case where one quark from each proton interacts via the exchange of a gluon. It is important to note that there are many possible diagrams for the scattering process, but we will draw a particular one. The diagram will have the following features: - Two incoming protons (labeled as \(p_1\) and \(p_2\)) with their quark content (\(u_1, u_2, d\) for \(p_1\) and \(u_3, u_4, d'\) for \(p_2\)). - The exchange of a gluon (labeled as \(g\)) between, for instance, \(u_1\) and \(u_3\). - Two outgoing protons (labeled as\(p_1'\) and \(p_2'\)) with their quark content (\(u_2, u_3', d\) for \(p_1'\) and \(u_4, u_1', d'\) for \(p_2'\)).

To draw the Feynman diagram for neutron beta decay, we need to consider the weak nuclear force mediated by W bosons. The process is characterized by the transition of a down quark in the neutron into an up quark, resulting in the transformation of a neutron into a proton. Simultaneously, an electron and an antineutrino are emitted. The diagram will have the following features: - An incoming neutron (labeled as \(n\)) and its quark content (\(u, d, d\)). - The emission of a W- boson (labeled as \(W^-\)) from the down quark. - The transformation of the down quark into an up quark and subsequent formation of a proton (labeled as \(p\)) and its quark content (\(u, u, d'\)). - The W- boson decays into an electron (labeled as \(e^-\)) and an electron antineutrino (labeled as \(\bar{\nu}_e\)).