Draw the lowest-order Feynman diagrams for the pair...

Draw the lowest-order Feynman diagrams for the pair annihilation process $$ \mathrm{e}^{+}\left(p_1, s_1\right)+\mathrm{e}^{-}\left(p_2, s_2\right) \rightarrow \gamma\left(k_1, \varepsilon_1\right)+\gamma\left(k_2, \varepsilon_2\right) $$ Check your answer with Fig. 6.13. Use the Feynman rules to show that $$ -i \mathscr{K}=(i e)^2 \bar{v}^{\left(s_1\right)}\left(p_1\right)\left(k_1^* \frac{i}{p_1-k_1-m} k_2^*+\xi_2^* \frac{i}{p_2-k_1-m} \xi_1^*\right) u^{\left(s_2\right)}\left(p_2\right) . $$ Verify that, in the high-energy limit, the spin-averaged rate is given by $$ \overline{|\mathscr{R}|^2}=2 e^4\left(\frac{u}{t}+\frac{t}{u}\right) \text {. } $$ The $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \gamma \gamma$ cross section has both forward and backward peaks, corresponding to the $t$ - and $u$-channel exchanged electrons being almost on mass shell. Result $(6.117)$ can also be obtained by crossing the amplitude for Compton scattering. To go from $\gamma \mathrm{e}^{-} \rightarrow \gamma \mathrm{e}^{-}$to $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \gamma \gamma$, we simple "cross" the ingoing photon with the outgoing electron, $$ \begin{aligned} & k, \varepsilon \rightarrow-k_2, \varepsilon_2^* \\ & p^{\prime} \rightarrow-p_1 \quad \text { and } \quad \bar{u}^{\left(s^{\prime}\right)}\left(p^{\prime}\right) \rightarrow \bar{v}^{\left(s_1\right)}\left(p_1\right) . \end{aligned} $$ The initial electron and outgoing photon are unaltered: $$ u^{(s)}(p) \equiv u^{\left(s_2\right)}\left(p_2\right), \quad k^{\prime}, \varepsilon^{\prime *} \rightarrow k_1, \varepsilon_1^* . $$ Making these substitutions in (6.103) and (6.104) gives the pair annihilation amplitude (6.117).

Draw the lowest-order Feynman diagrams for the pair annihilation process
$$
\mathrm{e}^{+}\left(p_1, s_1\right)+\mathrm{e}^{-}\left(p_2, s_2\right) \rightarrow \gamma\left(k_1, \varepsilon_1\right)+\gamma\left(k_2, \varepsilon_2\right)
$$
Check your answer with Fig. 6.13. Use the Feynman rules to show that
$$
-i \mathscr{K}=(i e)^2 \bar{v}^{\left(s_1\right)}\left(p_1\right)\left(k_1^* \frac{i}{p_1-k_1-m} k_2^*+\xi_2^* \frac{i}{p_2-k_1-m} \xi_1^*\right) u^{\left(s_2\right)}\left(p_2\right) .
$$

Verify that, in the high-energy limit, the spin-averaged rate is given by
$$
\overline{|\mathscr{R}|^2}=2 e^4\left(\frac{u}{t}+\frac{t}{u}\right) \text {. }
$$

The $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \gamma \gamma$ cross section has both forward and backward peaks, corresponding to the $t$ - and $u$-channel exchanged electrons being almost on mass shell. Result $(6.117)$ can also be obtained by crossing the amplitude for Compton scattering. To go from $\gamma \mathrm{e}^{-} \rightarrow \gamma \mathrm{e}^{-}$to $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \gamma \gamma$, we simple "cross" the ingoing photon with the outgoing electron,
$$
\begin{aligned}
& k, \varepsilon \rightarrow-k_2, \varepsilon_2^* \\
& p^{\prime} \rightarrow-p_1 \quad \text { and } \quad \bar{u}^{\left(s^{\prime}\right)}\left(p^{\prime}\right) \rightarrow \bar{v}^{\left(s_1\right)}\left(p_1\right) .
\end{aligned}
$$

The initial electron and outgoing photon are unaltered:
$$
u^{(s)}(p) \equiv u^{\left(s_2\right)}\left(p_2\right), \quad k^{\prime}, \varepsilon^{\prime *} \rightarrow k_1, \varepsilon_1^* .
$$

Making these substitutions in (6.103) and (6.104) gives the pair annihilation amplitude (6.117).

Key Concepts

Feynman Diagrams

Feynman diagrams are pictorial representations used in quantum field theory to describe and calculate the behavior of subatomic particles during interactions. They simplify complex mathematical expressions by mapping the physical process as vertices, internal lines (propagators), and external lines representing the incoming and outgoing particles. The diagrams indicate the order of the perturbation expansion and clearly show the exchanges of virtual particles during the interaction.

Feynman Rules

Feynman rules provide a systematic prescription for translating Feynman diagrams into mathematical expressions for scattering amplitudes. These rules specify how to assign mathematical factors to vertices, propagators, and external lines based on the underlying Lagrangian of the theory. They are essential for calculating the probability amplitudes and cross sections for various quantum processes.

Propagators

Propagators are elements of the Feynman diagrams that represent the evolution or the ‘propagation’ of virtual particles between interaction vertices. They include the momentum-dependent factors and the particle masses, encoding the behavior of intermediate states that are typically off mass shell. The propagator's structure plays a key role in determining the amplitude and singularity structure of the interaction.

Mandelstam Variables

Mandelstam variables (commonly denoted as s, t, and u) are Lorentz-invariant quantities used to describe the kinematics of scattering processes. They capture the energy and momentum transfer between particles and enable the analysis of scattering amplitudes in different channels. This framework facilitates comparison of processes and simplifies the evaluation of cross sections by expressing them in terms of these invariants.

Crossing Symmetry

Crossing symmetry is a profound concept in quantum field theory that relates different scattering processes by interchanging initial and final states. It allows one to derive the amplitude for one process from the amplitude of another, such as obtaining the pair annihilation amplitude from a Compton scattering amplitude. This symmetry reflects the underlying analytic properties of scattering amplitudes and reduces the amount of independent calculations required.

Spin Averaging and High-Energy Limit

Spin averaging is a technique used in calculating cross sections where the spin states of the particles are unobserved, leading to an average over initial spins and a sum over final spins. In the high-energy limit, simplifications occur because masses can often be neglected relative to the particle energies. This approach leads to expressions that expose the dominant kinematic dependencies, such as the appearance of Mandelstam variable ratios in the squared amplitude.

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