Connections between Deep-Inelastic and Annihilation Processes at Next-to-Next-to-Leading Order and Beyond

We have discovered 7 intimate connections between the published results for the radiative corrections, $\Ck$, to the Gross--Llewellyn Smith (GLS) sum rule, in deep-inelastic lepton scattering, and the radiative corrections, $\Cr$, to the Adler function of the flavour-singlet vector current, in $\ee$ annihilation. These include a surprising relation between the scheme-independent single-electron-loop contributions to the 4-loop QED $\beta$\/-function and the zero-fermion-loop abelian terms in the 3-loop GLS sum rule. The combined effect of all 7 relations is to give the factorization of the 2-loop $\beta$\/-function in \[\Ds\equiv\Ck\Cr-1=\frac{\Be}{\Aq}\left\{S_1\Cf\Aq+\left[S_2\Tf\Nf +\Sa\Ca+\Sf\Cf\right]\Cf\Aq^2\right\}+O(\Aq^4)\,,\] where $\Aq=\al(\mu^2=Q^2)/4\pi$ is the $\MS$ coupling of an arbitrary colour gauge theory, and \[S_1=-\Df{21}{2}+12\Ze3\,;\quad S_2=\Df{326}{3}-\Df{304}{3}\Ze3\,;\quad \Sa=-\Df{629}{2}+\Df{884}{3}\Ze3\,;\quad \Sf=\Df{397}{6}+136\Ze3-240\Ze5\] specify the sole content of $\Ck$ that is not already encoded in $\Cr$ and $\Be=Q^2\rd\Aq/\rd Q^2$ at $O(\Aq^3)$. The same result is obtained by combining the radiative corrections to Bjorken's polarized sum rule with those for the Adler function of the non-singlet axial current. We suggest possible origins of $\beta$ in the `Crewther discrepancy', $\Ds$, and determine $\Ds/(\Be/\Aq)$, to all orders in $\Nf\Aq$, in the large-$\Nf$ limit, obtaining the {\em entire\/} series of coefficients of which $S_1$ and $S_2$ are merely the first two members.