Abstract
Most primitively divergent Feynman diagrams are well defined in x-space
but too singular at short distances for transformation to p-space.
A new method of regularization is developed in which singular functions
are written as derivatives of less singular functions which contain
a logarithmic mass scale. The Fourier transform is then defined by
formal integration by parts. The procedure is extended to graphs
with divergent subgraphs. No explicit cutoff or counter-terms are
required, and the method automatically delivers renormalized amplitudes
which satisfy Callan-Symanzik equations. These features are thoroughly
explored in massless ?4 theory through 3-loop order, and the method
yields explicit functional forms for all amplitudes with less difficulty
than conventional methods which use dimensional regularization in
p-space. The procedure also appears to be compatible with gauge invariance
and the chiral structure of the standard model. This aspect is tested
in extensive 1-loop calculations which include the Ward identity
in quantum electrodynamics, the chiral anomaly, and the background
field algorithm in non-abelian gauge theories.
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