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A Feynman integral via higher normal functions. (English) Zbl 1365.81090

The authors picked up the so-called three banana scalar Feynman integral in two space-time dimensions to convert it into a subject of purely mathematical investigation. I mention for the uninitiated that Feynman integrals appear in the perturbative expansion of scattering amplitudes in quantum filed theories. If we go beyond the zeroth order approximation in this expansion, we will encounter the famous (and often divergent) Feynman integrals. The paper under consideration discusses the mathematical aspects of one such integral. In the first section the authors relate the integral to the third order Picard-Fuchs differential equation (with the variable being the external four momentum squared \(t\) ) and discuss in detail the inhomogeneous solution of this equation. The latter is given in terms of Bellinson-Levin elliptic trilogarthmic functions. The section finishes with the evaluation of the Feynman integral at special arguments, \(t=0\) and \(t=1\). The denominator of the integrand is associated wit \(K3\) surfaces with a specific Picard number. After a brief introduction to the subject in section three, the authors use the surface theory to relate the integral to higher normal functions which gives an alternative proof of the results obtained via the Picard-Fuchs differential equation. Yet another method to express the integral by elliptic trilogarithms is presented in section 5. The subsequent section treats the Feynman integral via the Hodge theory. This section ends with a discussion of the integral values at \(t=0\) and \(t=1\) which is not a repetition, but offers a different point of view on this matter. The cycle closes here. This brief description here of the paper shows what the authors intend to do: they try and (succeed) to make many inter-connections between different mathematical theories with the banana Feynman integral as the binding element.
The terminology of the paper is dense owing to the many inter-connections. One could also say that the pre-requisites are manifold. The reader should have heard about motivic cohomology, Dedekind eta function, Eisenstein series and symbols, L-Functions, the Milnor K-group, Higher Chow groups, Zariski closure, Deligne complex, Bellinson cycles, Gauss-Manin connection, Kuga variety and many other mathematical constructs and definitions. A physicist even acquainted with higher mathematics will wonder what has happened to the old Feynman integral, a mathematician might be delighted to see so many branches of mathematics united together around a physical object of interest.
The rather big paper is self contained (provided the reader’s diversity in mathematical education is vast enough) with proofs of the theorems and several discussions around them. The bibliography alone tells the unexpected: the banana Feynman integral has more ramifications than Feynman could imagine.

MSC:

81T18 Feynman diagrams
14D07 Variation of Hodge structures (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)

Citations:

Zbl 1365.81090
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References:

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