Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 2 / e / Solution

In $d=4-2ϵ$, the modified minimal subtraction scheme subtracts ${\Delta }_{\overline{\mathrm{MS}}}=1/ϵ-{\gamma }_E+\log (4\pi )$ for each one-loop pole, with no further finite, kinematics-dependent subtraction. Equivalently, use a rescaled renormalization scale ${\overline{\mu }}^2=4\pi e^{-{\gamma }_E}{\mu }_R^2$ and subtract only $1/ϵ$ in that convention. This differs from the minimal subtraction scheme by a fixed scale convention.

Let $C=e^2{\Delta }_{\overline{\mathrm{MS}}}/(16{\pi }^2)$. With the source's counterterm insertion signs, cancellation proceeds as follows. The vertex has $-ie{\gamma }^{\mu }C$ from the loop and $-ie{\gamma }^{\mu }{\delta }_1$ from its counterterm, giving ${\delta }_1=-C$. The kinetic part of the fermion self-energy insertion is $+i/pC$ and its counterterm is $+i/p{\delta }_2$, giving ${\delta }_2=-C$. Finally the photon vacuum polarization insertion is $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu })(4C/3)$ and its counterterm is $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu }){\delta }_3$, givingWriting $Z_i=1+{\delta }_i$ therefore verifies the Ward identity for QED renormalization constants, $Z_1=Z_2$, at one loop. The proper-vertex Ward-Takahashi identity in QED independently gives ${\Lambda }^{\mu }(p,p)=-\partial \Sigma (p)/\partial p_{\mu }$, whose divergent part reproduces the opposite signs of $F_1^{(1)}$ and $S$ found above. The equality holds in a common gauge fixing and subtraction prescription; the individual constants $Z_1,Z_2$ are gauge dependent.