Past exam of the gauge field theory course of the University of Cambridge / 2015 / Paper 1 / 3

At leading order in quantum electrodynamics, the electron-positron pair annihilates into one virtual photon, which produces the muon-antimuon pair. This is an s-channel Feynman diagram with $q=p_{-}+p_{+}=k_{-}+k_{+}$ and $q^2=s$. No QED vertex changes an electron directly into a muon, so there is no corresponding single-photon t-channel flavor-changing diagram.

Each charged-fermion vertex contributes $-ie{\gamma }^{\mu }$, the internal photon contributes $-i{g}_{\mu \nu }/(q^2+i0)$ in Feynman gauge, and external legs supply incoming $u(p_{-}),\bar{v}(p_{+})$ and outgoing $\bar{u}(k_{-}),v(k_{+})$. Their ordered contraction gives the stated amplitude. The overall convention-dependent phase does not affect its squared modulus.

For unpolarized scattering, average over the two spin choices of each incoming particle and sum over outgoing spins. The identities $\sum u\bar{u}=/p+m$ and $\sum v\bar{v}=/p-m$, together with ${\gamma }^0({\gamma }^{\mu }{)}^{†}{\gamma }^0={\gamma }^{\mu }$, convert the products into fermion spin sums:The factor $1/4$ averages the initial spins; there is no additional average over the final ones. In the massless limit the first trace is $4(p_{+\mu }{p}_{-\nu }+p_{+\nu }{p}_{-\mu }-g_{\mu \nu }{p}_{+}\cdot p_{-})$, with the analogous expression for the outgoing current. Contracting givesIn the centre-of-mass frame, every external energy is $E_{\mathrm{cm}}/2$. Choose $p_{-}$ along $+z$ and let $k_{-}$ make angle $\theta$ with it. The dot products are $p_{-}\cdot k_{-}=p_{+}\cdot k_{+}=s(1-\cos \theta )/4$ and $p_{-}\cdot k_{+}=p_{+}\cdot k_{-}=s(1+\cos \theta )/4$. HenceThe electron-positron annihilation into a muon pair cross-section is thereforeUsing $d\Omega =2\pi d(\cos \theta )$ givesIn ordinary units this becomes $4\pi {\alpha }^2(\hslash c{)}^2/(3E_{\mathrm{cm}}^2)$. It is the leading high-energy QED result: finite masses, radiative corrections and weak-interaction exchange modify the prediction outside that approximation.

Cross-section measurements can test spin assignments when the interaction and charges are specified. For example, keeping spin-half electrons but replacing the final muons by minimally coupled charged spin-zero particles replaces the fermion current by a momentum-difference current. In the massless limit, scalar-pair production by fermion annihilation gives $d\sigma /d\Omega ={\alpha }^2{\sin }^2\theta /(8s)$ and total $\pi {\alpha }^2/(3s)$. Its central angular peak differs from the fermionic $1+{\cos }^2\theta$ distribution.

Thus angular distributions, polarization information and suitably modeled rates distinguish spin hypotheses. A single total rate without assumptions about the couplings or the initial species does not independently establish both spins; changing interactions can mimic normalizations. The predicted fermion result is a testable consequence of the spin-half QED model, rather than a model-independent inference of spin from any one observed cross-section.