Dr. D.J. Broadhurst



Quantum chromodynamics

The following applications of the operator product expansion to QCD sum rules have been made: calculation of all non-perturbative contributions to light and heavy quark current correlations up to dimension d=6; phenomenology of charmonium and D decay; determination of the strange quark mass; development of a reliable method of minimal subtraction of mass singularities; calculation of the full Euler-Heisenberg lagrangian of QCD and of d=8 gluon condensate contributions to light-quark sum rules; two-loop gluon condensate radiative corrections; two-loop quark mass corrections and the absorption of their infra-red singularities by the quark condensate. An analysis of the renormalization of d=6 quark operators has been performed, in order to investigate a bound on anomalous dimensions and to analyze the absorption of perturbative mass singularities by condensates. The finite parts of two-loop quark mass and wave function renormalization have been calculated for the first time. The former is numerically important in the three-loop comparison of current and constituent quark masses; the latter is of importance in the heavy-quark effective field theory (HQET). Two-loop HQET anomalous dimensions have been calculated and applied in sum rules. A new relation between deep-inelastic and annihilation processes has been established. The matching of HQET and QCD currents has been achieved at two-loop order.

Renormalization of field theories

Boson-fermion cancellation of infinities in super-renormalizable field theories have been investigated up to 4 loops and necessary and sufficient conditions for completely finite three-dimensional theories have been systematically obtained and satisfied. Powerful new techniques for four-dimensional massless diagrams have been developed, leading to a result valid for all numbers of loops and to a method allowing the calculation of a large number of 2, 3, 4, and 5 loop diagrams contributing to propagators in scalar field theories. By exploiting conformal invariance, star-triangle duality and group theory, analytical results have been obtained for many diagrams previously thought intractable. A Z2 x S6 invariant expansion of the master two-loop diagram has been derived. This diagram has also been investigated by means of negative-dimensional integration. An infinite series of phi^3 diagrams has been evaluated and summed to yield the strong-coupling expansion. Powerful techniques for massive fields have yielded new analytical results by purely algebraic methods which exploit wreath product transformations. The 4-loop renormalization of QED has been performed, on-shell, with applications to the muon anomaly. Contributions to the Gell-Mann-Low function have been obtained to all orders. Construing renormalization as a skein operation on link diagrams that encode momentum flow, new connections between field theory, knot theory, and number theory, have been forged, and intensively investigated to 7-loop order. This has also given a better understanding of which Euler sums are irreducible.

Dirk Kreimer and I have been automating renormalization, using Hopf Algebra. Jon Borwein and I have been discovering other neat connections between everything and everything else, thanks to superb tools from people like David Bailey and Helaman Ferguson. If you are interested, take a look at:

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  • David Broadhurst, 25 August 2011