Natural renormalization

A careful analysis of differential renormalization shows that a distinguished choice of renormalization constants allows for a mathematically more fundamental interpretation of the scheme. With this set of a priori fixed integration constants differential renormalization is most closely related to the theory of generalized functions. The special properties of this scheme are illustrated by application to the toy example of a free massive bosonic theory. Then we apply the scheme to the phi^4-theory. The two-point function is calculated up to five loops. The renormalization group is analyzed, the beta-function and the anomalous dimension are calculated up to fourth and fifth order, respectively.Comment: 23 pages, LaTeX, AMSsymbols, epsf style, 3 PostScript figure

Geometries in perturbative quantum field theory

In perturbative quantum field theory one encounters certain, very specific geometries over the integers. These `perturbative quantum geometries' determine the number contents of the amplitude considered. In the article `Modular forms in quantum field theory' F. Brown and the author report on a first list of perturbative quantum geometries using the `$c_2$-invariant' in $\phi^4$ theory. A main tool was `denominator reduction' which allowed the authors to examine graphs up to loop order (first Betti number) 10. We introduce an improved `quadratic denominator reduction' which makes it possible to extend the previous results to loop order 11 (and partially orders 12 and 13). For comparison, also 'non-$\phi^4$' graphs are investigated. Here, we were able to extend the results from loop order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously 157)---while being consistent with all major $c_2$-conjectures---leads to a more refined picture of perturbative quantum geometries.Comment: 35 page

The geometry of one-loop amplitudes

We review a reduction formula by Petersson that reduces the calculation of a one-loop amplitude with N external lines in n<N space-time dimensions to the case n=N and give it a geometric interpretation. In the case n=N the calculation of the euclidean amplitude is shown to be equivalent to the calculation of the volume of a tetrahedron spanned by the momenta in (n-1)-dimensional hyperbolic space. The underlying geometry is intimately linked to the geometry of the reduction formula.Comment: 23 pages, 8 figure