Where do the tedious products of ζ’s come from?

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Where do the tedious products of ζ’s come from? (2002)

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by D. J. Broadhurst A

BibTeX

@MISC{A02wheredo,
    author = {D. J. Broadhurst A},
    title = {Where do the tedious products of ζ’s come from?},
    year = {2002}
}

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Abstract

Lamentably, the full analytical content of the ε–expansion of the master two–loop two–point function, with arbitrary self–energy insertions in 4 − 2ε dimensions, is still unknown. Here we show that multiple zeta values (MZVs) of weights up to 12 suffice through O(ε 9). Products of primitive MZVs are generated by a processes of “pseudo–exponentiation ” whose combinatorics faithfully accord with expectations based on Kreimer’s modified shuffle product and on the Drinfeld–Deligne conjecture. The existence of such a mechanism, relating thousands of complicated rational numbers, enables us to identify precise and simple combinations of MZVs specific to quantum field theories in even numbers of spacetime dimensions. 1. Master two–loop two–point function The object of study in this contribution is the two–loop integral I = p2(D/2−α6) πD ∫ ∫

Keyphrases

tedious product point function simple combination arbitrary self energy insertion pseudo exponentiation drinfeld deligne conjecture multiple zeta value full analytical content primitive mzvs complicated rational number shuffle product spacetime dimension field theory even number

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