Three-Loop Yang-Mills beta-Function via the Covariant Background Field   Method
  We demonstrate the effectivity of the covariant background field method by means of an explicit calculation of the 3-loop beta-function for a pure Yang-Mills theory. To maintain manifest background invariance throughout our calculation, we stay in coordinate space and treat the background field non-perturbatively. In this way the presence of a background field does not increase the number of vertices and leads to a relatively small number of vacuum graphs in the effective action. Restricting to a covariantly constant background field in Fock-Schwinger gauge permits explicit expansion of all quantum field propagators in powers of the field strength only. Hence, Feynman graphs are at most logarithmically divergent. At 2-loop order only a single Feynman graph without subdivergences needs to be calculated. At 3-loop order 24 graphs remain. Insisting on manifest background gauge invariance at all stages of a calculation is thus shown to be a major labor saving device. All calculations were performed with Mathematica in view of its superior pattern matching capabilities. Finally, we describe briefly the extension of such covariant methods to the case of supergravity theories. 
J.-P. Boernsen, Anton E. M. van de Ven