Use rotation matrix arguments to show that for "spinless"...

Use rotation matrix arguments to show that for "spinless" electrons and muons
$$
\mathscr{M}\left(\mathrm{e}^{-} \mathrm{e}^{+} \rightarrow \mu^{-} \mu^{+}\right) \propto \frac{t-u}{s}
$$
Compare the $s$-channel photon contribution of (4.47).

Key Concepts

Spinless Approximation

The spinless approximation assumes that the intrinsic spin degrees of freedom of the particles are ignored or treated as not contributing to the angular momentum dynamics. This simplification allows one to focus solely on orbital angular momentum effects and makes the derivation of the amplitude structure, such as the (t-u)/s factor, more tractable through the use of rotation matrix arguments.

s-Channel Photon Exchange

The s-channel photon contribution refers to the part of the scattering process mediated by a photon exchanged in the s-channel, where the photon is the virtual particle created in the annihilation of the electron-positron pair and then produces the muon-antimuon pair. Comparing this contribution involves examining how the photon propagator and its vertex factors lead to a similar dependence on Mandelstam variables as derived from the rotation matrix arguments.

Mandelstam Variables

Mandelstam variables (s, t, u) are Lorentz-invariant quantities that describe the kinematics of scattering processes. They encapsulate the energy and momentum transfer between particles and provide a convenient framework for expressing scattering amplitudes. Here, the dependence of the amplitude on (t-u)/s reflects the relative contributions of momentum and energy exchanges in the interaction.

Rotation Matrices

Rotation matrices are used to translate between different bases, especially when analyzing how quantum states transform under rotations. They are crucial in determining angular dependencies and symmetries in scattering processes. In this context, rotation matrices help establish the relationship between the angular momentum components of the initial and final states, providing insight into the form of the scattering amplitude.

Scattering Amplitude

The scattering amplitude quantifies the probability amplitude for a scattering event and is derived using the rules of quantum field theory. It encapsulates both the dynamics of the interaction and the symmetries of the process. The proportionality of the amplitude to (t-u)/s in this case arises from the interplay between the underlying Lorentz invariance and the transformation properties under rotations.

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