GKZ hypergeometric systems of the four-loop vacuum Feynman integrals

A bstract Basing on Mellin-Barnes representations and Miller’s transformation, we present the Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of 4-loop vacuum Feynman integrals with arbitrary masses. Through the GKZ hypergeometric systems, the analytical hypergeometric solutions of 4-loop vacuum Feynman integrals with arbitrary masses can be obtained in neighborhoods of origin including infinity. The analytical expressions of Feynman integrals can be formulated as a linear combination of the fundamental solution systems in certain convergent region, which the combination coefficients can be determined by the integral at some regular singularities, the Mellin-Barnes representation of the integral, or some mathematical methods.