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 Zelevinskiĭ theory on what are now commonly called GKZ-hypergeometric systems whose singularities are described by the principal A-determinant.

Tropical Feynman integration in the Minkowski regime

“…This means that conv( 𝐴) is not a matroid polytope so neither normality nor the GP property follows directly. From the results in [38,Corollary 5.6] it follows that N𝐴 is normal, however, conv( 𝐴) is not a GP.…”

“…• If no cancellation between F 0 and F 𝑚 occurs, 𝑝(𝑉 ′ ) 2 ≠ 0 for all 𝑉 ′ ⊂ 𝑉 ext and every internal vertex is connected to an external vertex via a massive path, then N[F ] is a GP and 𝐴 is normal [40] cf. [38].…”

“…• When 𝑚 𝑒 = 0 for all 𝑒 ∈ 𝐸 and 𝑝(𝑉 ′ ) 2 ≠ 0 for all 𝑉 ′ ⊂ 𝑉 ext , then N[F ] is a matroid polytope and thus GP and 𝐴 is normal [40] cf. [38].…”

Restrictions of Pfaffian systems for Feynman integrals

“…On the mathematical side, we believe that it would be interesting to investigate restriction ideals using the Macaulay matrix method. This would also aid in the computation of symbol alphabets at two loops, as they can be obtained from restrictions of GKZ systems [39].…”

“…We take our inspiration from a particularly well-studied holonomic D-module: the GKZ hypergeometric system [24] -though, as we show, the algorithms presented here also apply beyond this case. In the GKZ framework [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], one generalizes parametric representations of a Feynman integral to include extra variables, such that now z = (z 1 , . .…”