Draw all possible lowest-order Feynman diagrams for the processes: $$ \mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mu^{+} \mu^{-}, \quad \mathrm{e}^{+} \mathrm{e}^{-} \rightarrow v_{\mu} \bar{v}_{\mu}, \quad v_{\mu} \mathrm{e}^{-} \rightarrow v_{\mu} \mathrm{e}^{-} \quad \text { and } \quad \bar{v}_{\mathrm{e}} \mathrm{e}^{-} \rightarrow \bar{v}_{\mathrm{e}} \mathrm{e}^{-} . $$

Quantum field theory (QFT) is the mathematical framework in which we understand particle interactions at the subatomic level. It combines classical field theory, quantum mechanics, and special relativity. This theory helps predict how particles like electrons, muons, and neutrinos interact through fundamental forces.

One of QFT's key features is the use of Feynman diagrams, which visually represent particle interactions and simplify complex calculations. In these diagrams, particles are shown as lines, and their interactions are depicted as points called vertices.

Understanding QFT is crucial in analyzing the processes you're studying, such as electron-positron annihilation or neutrino-electron scattering. Each interaction can be represented using Feynman diagrams and follow certain rules derived from QFT principles.

In photon exchange interactions, particles exchange photons, which are the force carriers of the electromagnetic force. Photons are depicted by wavy lines in Feynman diagrams.

An example of photon exchange is the process \( e^+ e^- \rightarrow \mu^+ \mu^- \). In this process, an electron (\text{e}^-) and a positron (\text{e}^+) annihilate, producing a photon (\text{γ}). The photon then decays into a muon (\text{μ}^-) and an anti-muon (\text{μ}^+).

Photons are massless and travel at the speed of light. Their interaction is described by quantum electrodynamics (QED), a subset of quantum field theory that deals specifically with the electromagnetic force.

Z-boson exchange happens in electroweak interactions, which involve the exchange of Z-bosons (neutral force carriers) between particles. Z-bosons are represented by straight lines in Feynman diagrams.

An example is the process \( e^+ e^- \rightarrow \mu^+ \mu^- \). Besides photon exchange, this process can also occur through Z-boson exchange. In this case, an electron and a positron annihilate to produce a Z-boson (\text{Z}). The Z-boson then decays into a muon and an anti-muon.

Another instance of Z-boson exchange is the process \( v_{\mu} e^- \rightarrow v_{\mu} e^- \). Here, a muon neutrino (\text{v}_{\text{μ}}) and an electron (\text{e}^-) interact and scatter via an exchanged Z-boson. This exchange illustrates the weak interaction, which operates at a subatomic level and is closely related to the electromagnetic force.

Particle interactions are fundamental events where particles interact and exchange force carriers. Depending on the forces involved, these carriers can be photons, Z-bosons, or others.

In the exercises you reviewed, different interactions come into play. For instance:

• \( e^+ e^- \rightarrow \mu^+ \mu^- \): This can occur via photon or Z-boson exchange.
• \( e^+ e^- \rightarrow v_{\mu} \bar{v}_{\text{μ}} \): This mainly involves Z-boson exchange because neutrinos interact through the weak force.
• \( v_{\mu} e^- \rightarrow v_{\mu} e^- \): Represents neutrino-electron scattering, best described through Z-boson exchange.
• \( \bar{v}_{\text{e}} e^- \rightarrow \bar{v}_{\text{e}} e^- \): Similarly, this interaction involves Z-boson exchange.

Understanding these interactions helps in visualizing and analyzing how fundamental particles interact to form the universe's structure.