Fourier transformation sends $\partial_\mu$ to $iq_\mu$. Write the connected current three-point function as the two full <Dirac propagators> joined to the amputated vertex:
$$
\langle j^\mu\psi\bar\psi\rangle_{m conn}
=G(p_1)V_3^\mu(q,p_1,p_2)G(p_2),
\qquad q=p_1-p_2.
$$
The two contact terms remove one propagator at a time. Multiplying the transformed identity by $G^{-1}(p_1)$ on the left and $G^{-1}(p_2)$ on the right yields
$$
\boxed{iq_\mu V_3^\mu(q,p_1,p_2)
=ie\bigl[G^{-1}(p_1)-G^{-1}(p_2)\bigr]},
\qquad p_2=p_1-q.
$$
This is the momentum-space <Ward-Takahashi identity> for the full vertex and full propagator.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-304/3/c/solution