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

The classical equations areFor any time-ordered functional $\mathcal{O}$, the three Schwinger-Dyson equations state that insertion of each left-hand side equals the corresponding contact terms obtained by $-i$ times the functional derivative of $\mathcal{O}$ with respect to $phi$, $\bar{\psi }$, or $\psi$, with the usual reversed order and Grassmann signs for fermionic derivatives.

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 $(2\pi {)}^4{\delta }^{(4)}(k-p-q)$.

Iterating the Schwinger-Dyson equations once, the leading connected correlator is one scalar propagator and two Dirac propagators meeting at one Yukawa interaction vertex. Amputation givesThus the invariant amplitude, in the convention $S_{fi}=i(2\pi {)}^4{\delta }^{(4)}{\mathcal{M}}_{fi}$, is ${\mathcal{M}}_{s,s^{\prime }}=-\lambda {\bar{u}}^s(p)v^{s^{\prime }}(q)$.

The spin sum becomes a gamma-matrix trace:Since $(p+q{)}^2=M^2$, $p\cdot q=(M^2-2m^2)/2$. Therefore the normalization requested in the paper gives

The pseudoscalar vertex replaces $-i\lambda$ by $g{\gamma }^5$, soUsing ${\gamma }^5/q{\gamma }^5=-/q$ with the conjugation sign givesThe interactions are experimentally distinguishable through the decay rate and its threshold behavior: scalar decay is proportional to $(1-4m^2/M^2{)}^{3/2}$, whereas pseudoscalar decay is proportional to $(1-4m^2/M^2{)}^{1/2}$. Spin and parity correlations provide further discrimination.