Large-order Behaviour of the QCD Adler D-function in Planar Approximation

We consider the structure of the leading ultra-violet (UV) renormalon singularity associated with the QCD vacuum polarization Adler D-function, in the approximation that only planar Feynman diagrams are retained. This ``planar approximation’’ results in some simplifications, in particular three of the four potential contributions from four-fermion operators are shown to be absent. We are able to obtain a fully normalized result for the leading $n\to \mathrm{\infty }$ behaviour of the portion of perturbative coefficients proportional to $N_f^{n-r}{N}^r$, for SU(N) QCD with $N_f$ quark flavours.