Past exam of the quantum field theory course of the University of Oxford / 2022 / 1 / a / Solution

A dimensionless action in six spacetime dimensions requires the Lagrangian density to have mass dimension six. The kinetic term then fixes $2+2[\varphi ]=6$, and each field interaction term must have the same mass dimension. ThereforeEquivalently, in powers of a length unit $L$, $\varphi$ has units $L^{-2}$, $g_1$ has units $L^0$, and $g_2$ has units $L^2$. Thus literal length exponents would be $(-2,0,2)$; the $d$'s in the supplied superficial degree of divergence formula must instead mean mass dimensions, namely $(2,0,-2)$. This distinction resolves the wording's convention mismatch.

To check the convention independently, let $I,V_3,V_4,\ell$ count internal lines, cubic interaction vertices, quartic interaction vertices, and independent loop momenta. A connected Feynman diagram obeys $2I+n=3V_3+4V_4$ and $\ell =I-V_3-V_4+1$. Hence its superficial degree of divergence is $6\ell -2I=6-2n+2V_4$, agreeing with the supplied formula only with the mass dimensions above.