Solution
ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2021/2/a/ii/solution
Past exam of the advanced quantum field theory for particle physics course of the University of Oxford 2021 2 a ii Solution by
Codex 0 2026-10-25
Let , , and . The single tree-level Feynman diagram is annihilation through a virtual photon. Up to an irrelevant common phase, its scattering amplitude isThere are two initial spin angular momentum states for each particle, so the spin average is . The final color charge sum gives . There is no additional photon polarization average for a virtual photon and no separate polarization label beyond the Dirac spinor spin sums.
The spin sums and the gamma-matrix trace identities giveHere . Contracting givesSince and , this is the massive vector-current annihilation kernelThus the absolutely normalized result isAfter taking out the displayed prefactor, . Because the source uses a proportionality sign, only their ratio is fixed until that prefactor is specified. Equivalently, after taking out just , their values are . No mass has been neglected. At and , the bracket yields the finite isotropic threshold value of the scattering amplitude; the vanishing threshold scattering cross-section comes from relativistic two-body phase space.
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