Abstract
Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel hat{Gamma}hbox{-} mathrm{class} evaluation in the ambient spaces of the mirror, while the imaginary part of the integral in this regime is determined by the hat{Gamma}hbox{-} mathrm{class} of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the integrals when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogeneous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the banana integral in very short time to very high numerical precision for all values of the physical parameters. Using modular properties of the periods we determine the value of the maximal cut equal mass four-loop integral at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.
Highlights
B t loop orders in the perturbative expansion
Scalar Feynman integrals satisfy in dimensional regularization integration-by-parts identities [1, 2]1 (IBP) which allow to further reduce to a smaller number of integrals, commonly called master integrals of the respective problem. The latter are Feynman integrals associated with subtopologies in the above sense and in this way banana type integrals2 frequently arise — for instance as master integrals in two-loop electro-weak computations [6], in the two-loop Higgs+jet production cross section [7], in three-loop corrections to the ρ-parameter [8] or at four-loop order in the anomalous magnetic moment of the electron [9]
In the cohomology of a family of Calabi-Yau (l − 1)-folds, and the maximal cut integral of the Feynman graph turns out to be a special case thereof.8. In practice this means that the Feynman integral will be a linear combination of the homogeneous solutions and a special inhomogeneous solution
Summary
We introduce the main object we focus on in this paper, namely the l-loop banana Feynman integral and make first comments on the underlying geometry. We give a representation of the Feynman integral in terms of Bessel functions valid for small momenta. In the large momenta regime we calculate the maximal cut integral
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