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Using the Gelfand-Kapranov-Zelevinsk\u{\i} system for the primitivecohomology of an infinite series of complete intersection Calabi-Yau manifolds,whose dimension is the loop order minus one, we completely clarify the analyticstructure of all banana amplitudes with arbitrary masses. In particular, wefind that the leading logarithmic structure in the high energy regime, whichcorresponds to the point of maximal unipotent monodromy, is determined by anovel $\widehat \Gamma$-class evaluation in the ambient spaces of the mirror,while the imaginary part of the amplitude in this regime is determined by the$\widehat \Gamma$-class of the mirror Calabi-Yau manifold itself. We providesimple closed all loop formulas for the former as well as for the Frobenius$\kappa$-constants, which determine the behaviour of the amplitudes, when themomentum 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 thelatter case a complete set of (inhomogenous) Picard-Fuchs differentialequations for arbitrary masses. This allows to evaluate the amplitude as wellas other master integrals with raised powers of the propagators in very shorttime to very high numerical precision for all values of the physicalparameters. Using a recent $p$-adic analysis of the periods we determine thevalue of the maximal cut equal mass four-loop amplitude at the attractor pointsin terms of periods of modular weight two and four Hecke eigenforms and thequasiperiods of their meromorphic cousins.

Analytic Structure of all Loop Banana Amplitudes