Use the Feynman rules to evaluate ℛ(γe^- →γe^-)corresponding to the two Feynman diagrams of Fig.

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Use the Feynman rules to evaluate ℛ(γe^- →γe^-)corresponding...

Use the Feynman rules to evaluate $\mathscr{R}\left(\gamma \mathrm{e}^{-} \rightarrow \gamma \mathrm{e}^{-}\right)$corresponding to the two Feynman diagrams of Fig. 6.12 with the electron taken to have spin 0 . Show that the result is not invariant under the gauge transformation (6.66). Demonstrate that gauge invariance is restored if diagram 6.16 is included with a vertex factor $2 i e^2 \mathrm{~g}^{\mu \nu}$.

Key Concepts

Contact Interaction

A contact interaction refers to a term in the Lagrangian or an additional Feynman diagram that corresponds to an instantaneous interaction, typically represented by a vertex without any associated propagator. In gauge theories, such contact terms can play a crucial role in restoring gauge invariance by canceling gauge-variant parts arising from other diagrams, as is seen when a vertex factor proportional to the metric tensor is included.

Feynman Rules

Feynman rules provide a systematic prescription for translating the graphical representation of particle interactions in Feynman diagrams into precise mathematical expressions. These rules assign propagators to internal lines, vertex factors to interaction points, and external factors to incoming and outgoing particles, allowing for the calculation of scattering amplitudes within the framework of quantum field theory.

Scattering Amplitudes

The scattering amplitude is a complex quantity computed from a set of Feynman diagrams that encapsulates the probability amplitude for a specific process to occur. Evaluating the amplitude involves summing contributions from different diagrams, each corresponding to a particular physical process, and plays a central role in predicting measurable cross sections in high-energy physics experiments.

Gauge Invariance

Gauge invariance is a fundamental symmetry principle stating that physical observables should remain unchanged under local transformations of the fields. This symmetry ensures the consistency, renormalizability, and unitarity of the theory. In the context of quantum electrodynamics, maintaining gauge invariance means that the calculated scattering amplitudes must not depend on the arbitrary choices made in the gauge-fixing procedure.

Gauge Transformation

A gauge transformation alters the fields in a quantum field theory without changing the physical content, reflecting the redundancy in field descriptions inherent to gauge theories. The scattering amplitudes are expected to be invariant under such transformations; however, certain Feynman diagrams calculated in isolation may not exhibit this invariance. This noninvariance signals the necessity of including additional contributions that balance the gauge noninvariant pieces.

Scalar QED

Scalar quantum electrodynamics (QED) is a variation of the standard formulation where the charged matter field is taken to be a spin-zero field rather than a spin-½ field. This simplification, such as considering a scalar electron, allows for a clearer examination of the interplay between interaction vertices and gauge invariance without the complications introduced by spinor structures, while still maintaining the essential features of electromagnetic interactions.

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