We have discovered 7 intimate connections between the published results for the radiative corrections, C K , to the Gross-Llewellyn Smith (GLS) sum rule, in deepinelastic lepton scattering, and the radiative corrections, C R , to the Adler function of the flavour-singlet vector current, in e + e − annihilation. These include a surprising relation between the scheme-independent single-electron-loop contributions to the 4-loop QED β-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 β-function in INR-820/93 OUT-4102-45 1 ) In memoriam Sergei Grigorievich Gorishny, 1958-1988 2 ) D.Broadhurst@open.ac.uk 3 ) Kataev@inucres.msk.su

In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and small q 2 , in d dimensions, using identities derived from integration by parts. Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations. Padé approximants to truncations of the expansions are shown to be of great utility. As an application, we obtain the two-loop photon selfenergy, for all d, and achieve highly accelerated convergence of its expansions in powers of q 2 /m 2 or m 2 /q 2 , for d = 4.

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The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, H R , generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra H ladder of pure ladder diagrams and the Connes-Moscovici noncocommutative subalgebra H CM of noncommutative geometry. These three Hopf algebras admit a bigrading by n, the number of nodes, and an index k that specifies the degree of primitivity. In each case, we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainder. The results for H ladder are familiar from the theory of partitions, while those for H CM involve novel transforms of partitions. Most beautiful is the bigrading of H R , the largest of the three. Thanks to Sloane's superseeker, we discovered that it saturates all possible inequalities. We prove this by using the universal Hochschild-closed one-cocycle B + , which plugs one set of divergences into another, and by generalizing the concept of natural growth beyond that entailed by the Connes-Moscovici case. We emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory.

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