Symbol Alphabets from the Landau Singular Locus

Abstract

We provide evidence through two loops, that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the principal $A$-determinant. Focusing on one loop, we further show that all square-root letters may also be obtained, by re-factorizing the principal $A$-determinant with the help of Jacobi identities. We verify our findings by explicitly constructing canonical differential equations for the one-loop integrals in both odd and even dimensions of loop momenta, also finding agreement with earlier results in the literature for the latter case. We provide a computer implementation of our results for the principal $A$-determinants, symbol alphabets and canonical differential equations in an accompanying Mathematica file. Finally, we study the question of when a one-loop integral satisfies the Cohen-Macaulay property and show that for almost all choices of kinematics the Cohen-Macaulay property holds. Throughout, in our approach to Feynman integrals, we make extensive use of the Gel'fand, Graev, Kapranov and Zelevinski\u{\i} theory on what are now commonly called GKZ-hypergeometric systems whose singularities are described by the principal $A$-determinant.