Related papers: On an approach to canonicalizing elliptic Feynman …
We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse…
Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter $N$ or continuous parameter $x$. The analysis of these equations reveals to which order they…
A cosmological polytope is defined for a given Feynman diagram, and its canonical form may be used to compute the contribution of the Feynman diagram to the wavefunction of certain cosmological models. Given a subdivision of a polytope, its…
We introduce the regularized integrals for decorated graphs on elliptic curves, which produces an almost holomorphic function on upper half plane. Then we give the graph version of holomorphic anomaly equation to study the anti-holomorphic…
Symbol letters are crucial for analytically calculating Feynman integrals in terms of iterated integrals. We present a novel method to construct the symbol letters for a given integral family without prior knowledge of the canonical…
The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is…
We derive useful reduction formulae which express one-loop Feynman integrals with a large number of external momenta in terms of lower-point integrals carrying easily derivable kinematic coefficients which are symmetric in the external…
We introduce two novel numerical approaches for computing Feynman integrals based on their complete monotonicity (CM) and Stieltjes properties. The first method uses that scalar Feynman integrals are CM, meaning that all their derivatives…
We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of…
The method of differential equations in canonical form has proven a powerful tool for solving multiloop Feynman integrals. In this note we test this procedure away from four dimensions. Namely, we consider the simple example of a massless…
Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct $d\log$-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals,…
The integration of differential equations of Feynman integrals can be greatly facilitated by using a canonical basis. This paper presents the Mathematica package CANONICA, which implements a recently developed algorithm to automatize the…
The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to $d$-dimensional $d\log$-form…
The aim of these notes is to describe how to construct canonical bundles of moving frames and differential invariants for parametrized curves in Lagrangian Grassmannians, at least in the monotonic case. Such curves appear as Jacobi curves…
Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can…
We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…
The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation…
The evaluation of multi-loop Feynman integrals is one of the main challenges in the computation of precise theoretical predictions for the cross sections measured at the LHC. In recent years, the method of differential equations has proven…
Feynman's path integral is herein generalized to the nonextensive canonical density matrix based on Tsallis entropy. This generalization is done in two ways by using unnormalized and normalized constraints. Firstly, we consider the path…
Lusztig defined certain involutions on the equivariant K-theory of Slodowy varieties and gave a characterization of certain bases called canonical bases. In this paper, we give a conjectural generalization of these involutions and…