Lorentz-covariant rank-two loop reduction ($\int d^{d}q{q}_{\mu }{q}_{\nu }f(q^2)=\frac{g_{\mu \nu }}{d}\int d^{d}q{q}^{2}f(q^2)$)

With an invariant integration measure and no external vector in $f(q^2)$, the rank-two integral must be proportional to the metric. Contracting its indices fixes the coefficient to $1/d$. In dimensional regularization this dimension is the continued $d$, rather than four; retaining its dependence matters when multiplying a divergent loop integral.