Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 3 / a / ii / Solution

Define asymptotic states $|\varphi (k)⟩=a_k^{†}|0⟩$ and $|\psi (p,s)\bar{\psi }(q,s^{\prime })⟩=b_{p,s}^{†}{c}_{q,s^{\prime }}^{†}|0⟩$. The LSZ reduction formula reduces the decay matrix element from the connected three-point functionIn momentum space, amputate the scalar leg with $i(k^2-M^2)$ and the fermion legs with the inverse Dirac propagators, contract them with ${\bar{u}}^s(p)$ and $v^{s^{\prime }}(q)$, and take $k^2=M^2$, $p^2=q^2=m^2$. The spacetime integrations produce $(2\pi {)}^4{\delta }^{(4)}(k-p-q)$.