Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2017 / 2 / a

Use the source term $\int d^{4}x[J\varphi +\bar{\varsigma }\psi +\bar{\psi }\varsigma ]$ and left Grassmann functional derivatives. Differentiating its exponential inserts fields according toThe sign for $\bar{\psi }$ follows because a left derivative with respect to $\varsigma$ must cross $\bar{\psi }$. Keep the order $\bar{\psi }{\gamma }_5\psi \varphi$ when replacing these fields. Expand $e^{i{S}_{\mathrm{int}}}$ in the original path integral; each field monomial can now be generated by these ordered derivatives acting on the free functional. Resumming givesThe arguments refer respectively to $\varphi ,\bar{\psi },\psi$ at $x$. The free functional arises by performing the free Gaussian field integrals, and $\mathcal{N}$ may be chosen so that $Z[0]=1$.

No additional field-integration measure remains in this differential-operator formula. Such a measure is printed outside the operator in the source even though $Z_0$ has already integrated out those fields. At $\lambda =0$, that literal expression would contain $\int \mathcal{D}\varphi \mathcal{D}\bar{\psi }\mathcal{D}\psi 1$ rather than simply $Z_0$, which is not the free generating functional. Also, when ${\mathcal{L}}_{\mathrm{int}}$ denotes a density, the spacetime integral in the exponent is required.

At first order the Dyson expansion inserts $i{S}_{\mathrm{int}}=\lambda \int d^{4}x{\bar{\psi }}_{\alpha }(x)({\gamma }_5{)}_{\alpha \beta }{\psi }_{\beta }(x)\varphi (x)$. Contract the scalar field at the vertex with the external scalar. The four remaining fermion fields have two admissible pairings: both external fermions attach to the vertex, or the two vertex fermions contract with each other while the external pair contracts freely. Wick's theorem givesThe minus sign in the second term is the fermion sign of the closed contraction: $⟨T{\bar{\psi }}_{\alpha }(x){\psi }_{\beta }(x)⟩=-i{S}_F(0{)}_{\beta \alpha }$. The first term is connected; the second is a scalar-attached fermion tadpole multiplied by a disconnected free external fermion propagator. The full time-ordered expectation includes both topologies, whereas the connected generating functional retains only the first. There is no order-${\lambda }^2$ contribution in this pure Yukawa expansion because the total number of scalar fields would be odd.

For a parity-preserving free propagator, the tadpole coefficient is $\mathrm{tr}[{\gamma }_5{S}_F(0)]$, whose integrand contains only $\mathrm{tr}{\gamma }_5$ and $\mathrm{tr}({\gamma }_5{\gamma }^{\mu })$. Both vanish. Thus the second term is a legitimate formal Wick contraction but is zero with a compatible regulator; normal ordering the interaction removes it from the outset.