Hyperfunction approach to perturbative amplitudes.

Recently several inconsistencies between different measurements of hadronic physics sensitive quantities are getting prominence at a few sigma level. These include proton radius and muon g-2 measurements. Interpretation of data for these experiments requires careful analysis of higher order QED radiative corrections. These calculations are very challenging and there currently exists no reliable method to evaluate these RCs at second order and beyond. We report on the hyperfunction (HF) approach to this problem. We analyze deep algebro-geometric and number theoretic properties of loop integrals. This includes the analysis of type of singularities and their locus of the analytic continuation of the amplitudes. Among the results is the method to analyze the singularity locus, its singularities (ingeometric sense - as degenerations of in the normal bundle), and intersection theory of its components. The analytically continued amplitude forms a vector bundle on an open complex manifold. We discuss  methods used to obtain flat connections on this bundle , putting the usual master integrals approach into the framework of toric geometry. We discuss the use of generalized hypergeometric functions to construct convergent series for amplitudes. We mention the relation of the amplitude calculation with multidimensional isomonodromy problem for holonomic D-modules with Fuchsian singularities.

EIC experiments will provide exciting opportunities to study the partonic structure inside the proton and other hadrons. The interpretation of the experimental data (such as subthreshold production of phi and J/psi, SIDIS measurements, nucleon tomography etc.) crucially relies on QCD factorization, which itself relies on analysis of Landau singularities in perturbation theory. Our approach provides a deeper insight into the geometry of these objects and hopefully will shed light on the possible ways to build non-perturbative QCD formalism.