Abstract
We integrate three-loop sunrise-type vacuum diagrams in D0=4 dimensions with four different masses using configuration space techniques. The finite parts of our results are in numerical agreement with corresponding three-loop calculations in momentum space. Using some of the closed form results of the momentum space calculation we arrive at new integral identities involving truncated integrals of products of Bessel functions. For the non-degenerate finite two-loop sunrise-type vacuum diagram in D0=2 dimensions we make use of the known closed form p-space result to express the moment of a product of three Bessel functions in terms of a sum of Clausen polylogarithms. Using results for the nondegenerate two-loop sunrise diagram from the literature in D0=2 dimensions we obtain a Bessel function integral identity in terms of elliptic functions.
Highlights
The computation of higher order vacuum diagrams in quantum field theory has been extended to ever higher loop levels involving different degrees of mass degeneracy and/or zero mass values for their mass configurations
The comparison of the results of the two calculations will provide a welcome nontrivial cross check on the correctness of the respective p- and x-space calculations in as much as a subclass of the vacuum diagrams are of sunrise-type
We have recalculated nondegenerate sunrise-type three-loop vacuum integrals in x-space and have found numerical agreement with corresponding results obtained by p-space calculations
Summary
The computation of higher order vacuum diagrams in quantum field theory has been extended to ever higher loop levels involving different degrees of mass degeneracy and/or zero mass values for their mass configurations. If one can avail of closed form p-space results, one arrives at new nontrivial integral identities involving moments of Bessel functions. The particular form of the splitting technique preserves the inherent symmetry of the nondegenerate multiloop vacuum and sunrise integrals w.r.t. the exchange of different rungs or masses in the diagrams. This leads to the notion of truncated Bessel integrals.
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