Abstract
We evaluate the three-loop massive vacuum bubble diagrams in terms of polylogarithms up to weight six. We also construct the basis of irrational constants being harmonic polylgarithms of arguments ekiπ/3.
Highlights
Algebra of the sixth root of unity
We evaluate the three-loop massive vacuum bubble diagrams in terms of polylogarithms up to weight six
We proceed by studying three-loop vacuum integrals with a single mass scale at weights five and six
Summary
The integer number w is called the weight of the polylogarithm. The complete basis of the algebra of the sixth root of unity Aω through weight 6 has recently been constructed in ref. We construct the basis of the subalgebra of Aω formed by the harmonic polylogarithms [28] Hn1...np(z) of arguments zk = ωk. We shall call such an algebra AH(ωk). The numbers of the basis elements at each weight for the algebra of the sixth root of unity Aω and for AH(ω) are summarized in table 1. We apply the constructed bases Re Hw and Im Hw to the evaluation of the three-loop massive vacuum bubble diagrams
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