Progress with large Nf β-functions

We report on the progress in the computation of the β-functions of Φ4 theory and QCD in the large N expansion. For the former we give an analytic formula for the critical exponent which encodes higher order coefficients in the series beyond those currently known in the MS scheme at O(1N2) in a large N expansion. This allows us to deduce new coefficients in the field anomalous dimension which relate to Kreimerʹs knot theory of the divergences of a quantum field theory. A similar exponent is computed for the β-function of QCD at O(1Nf).