Related papers: Reduction to master integrals and transverse integ…
The reduction of Feynman integrals to a basis of master integrals plays a crucial role for many high-precision calculations and Kira is one of the leading tools for this task. In these proceedings we discuss some of the new features and…
This paper is concerned with algorithms and applications of decreasing minimization on an M-convex set, which is the set of integral elements of an integral base-polyhedron. Based on a recent characterization of decreasingly minimal…
We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master…
We provide new theoretical results in the field of inverse regression methods for dimension reduction. Our approach is based on the study of some empirical processes that lie close to a certain dimension reduction subspace, called the…
Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break…
Starting from basic identities of the group E8, we perform progressive reductions, namely decompositions with respect to the maximal and symmetric embeddings of E7xSU(2) and then of E6xU(1). This procedure provides a systematic approach to…
We prove a new identity for Feynman periods that acts on five-vertex cuts of completed primitive Feynman graphs. It is shown that in $\phi^4$ theory this identity is independent from existing identities which are the twist, the Fourier…
We compute the full set of two-loop Feynman integrals appearing in massless two-loop four-point functions with two off-shell legs with the same invariant mass. These integrals allow to determine the two-loop corrections to the amplitudes…
We present a new method for the numerical evaluation of loop integrals which is based on the Feynman Tree Theorem. The loop integrals are replaced by phase-space integration over fictitious extra on-shell particles. This integration can be…
In this thesis, we study the three-loop QCD form factors. After an introduction and a discussion of the physics motivation, we generate the quark form factor using Qgraf. We then show how to bring the Feynman integrals into a unique form by…
Modeling of multidimensional signal using tensor is more convincing than representing it as a collection of matrices. The tensor based approaches can explore the abundant spatial and temporal structures of the mutlidimensional signal. The…
We present a new algorithm for the reduction of one-loop tensor Feynman integrals within the framework of the XLOOPS project, covering both mathematical and programming aspects. The new algorithm supplies a clean way to reduce the one-loop…
We review the Laporta algorithm for the reduction of scalar integrals to the master integrals and the differential equations technique for their evaluation. We discuss the use of the basis of harmonic polylogarithms for the analytical…
We study the algebraic and analytic structure of Feynman integrals by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour. This operation is a coaction. It…
In this work, we introduce a new generalized integral transform involving many potentially known or new transforms as special cases. Basic properties of the new integral transform, that investigated in this work, include the existence…
The operator approach to analytical evaluation of multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of massless Feynman integrals, such as the integration by parts method and the method of…
In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and $\varepsilon$-factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric…
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…
We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis.
New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts…