Related papers: Improving integration-by-parts and differential eq…
We present a prescription for choosing orthogonal bases of differential $n$-forms belonging to quadratic twisted period integrals, with respect to the intersection number inner product. To evaluate these inner products, we additionally…
By introducing an auxiliary parameter, we find a new representation for Feynman integrals, which defines a Feynman integral by analytical continuation of a series containing only vacuum integrals. The new representation therefore…
A new powerful method to calculate Feynman diagrams is proposed. It consists in setting up a Taylor series expansion in the external momenta squared (in general multivariable). The Taylor coefficients are obtained from the original diagram…
This course on Feynman integrals starts from the basics, requiring only knowledge from special relativity and undergraduate mathematics. Topics from quantum field theory and advanced mathematics are introduced as they are needed. The course…
The universal method of expansion of integrals is suggested. It allows in particular to derive the threshold expansion of Feynman integrals.
Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This…
The Feynman checkerboard problem is an interesting path integral approach to the Dirac equation in `1+1' dimensions. I compare two approaches reported in the literature and show how they may be reconciled. Some physical insights may be…
It is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is…
We show how information on the uniformity properties of a point set employed in numerical multidimensional integration can be used to improve the error estimate over the usual Monte Carlo one. We introduce a new measure of (non-)uniformity…
We provide a new method to calculate the full microlocal description of singularities of Feynman integrals. This is done by associating a unique constructible function to the system of partial differential equations (PDEs) annihilating the…
Parametric Feynman integrals with the regions of integration defined by some polynomials are considered in this paper. It is shown that integrals with irregular integration regions can be converted to standard parametric integrals, for…
We present a novel approach for loop integral reduction in the Feynman parametrization using intersection theory and relative cohomology. In this framework, Feynman integrals correspond to boundary-supported differential forms in the…
Since Feynman integrals (FIs) at higher spacetime dimensions are free of infrared and collinear divergence--and their ultraviolet divergences can be systematically subtracted--this allows us to construct a wide range of locally finite…
We describe how a dlog representation of Feynman integrals leads to simple differential equations. We derive these differential equations directly in loop momentum or embedding space making use of a localization trick and generalized…
In a recent paper \cite{ft} a new powerful method to calculate Feynman diagrams was proposed. It consists in setting up a Taylor series expansion in the external momenta squared. The Taylor coefficients are obtained from the original…
The worldline formalism allows one to obtain compact integral representations combining the information of large numbers of Feynman diagrams. However, their analytic calculation leads to a non-standard integration problem for which existing…
Feynman integral reduction based on intersection theory provides an alternative to the traditional integration-by-parts method, yet its practical application has been constrained by the large number of variables required in the computation.…
A short review of expansion by regions is presented. It is a well-known strategy to obtain an expansion of a given multiloop Feynman integral in a given limit where some kinematic invariants and/or masses have certain scaling measured in…
Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals,…
In this paper, we describe a numerical approach to evaluate Feynman loop integrals. In this approach the key technique is a combination of a numerical integration method and a numerical extrapolation method. Since the computation is carried…