Related papers: Explicit solutions of the 3--loop vacuum integral …
We derive recurrence relations between phase space expressions in different dimensions by confining some of the coordinates to tori or spheres of radius $R$ and taking the limit as $R \to \infty$. These relations take the form of mass…
In a previous paper a new approach has been introduced for computing, recursively and numerically, one-loop tensor integrals. Here we describe a few modifications of the original method that allow a more efficient numerical implementation…
We use elementary methods to establish three key recurrence relations: one for derangement numbers, a second for harmonic numbers, and a third for degenerate harmonic numbers. Our results not only contribute to the understanding of the…
Recent results and methods of three-loop calculations in HQET are reviewed.
New analytic formulas for one-loop three-point Feynman integrals in general space-time dimension ($d$) are presented in this paper. The calculations are performed at general configurations for internal masses and external momenta. The…
We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.
The equations for a self-similar solution of an inviscid incompressible fluid are mapped into an integral equation which hopefully can be solved by iteration. It is argued that the exponent of the similarity are ruled by Kelvin's theorem of…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
New solution method for the systems of linear equations in commutative integral domains is proposed. Its complexity is the same that the complexity of the matrix multiplication.
The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be…
The paper is devoted to the methods of solving simultaneous recurrences. Specifically, we discuss transformation of matrix recurrences to regular recurrences and propose a way of solving special matrix recurrences of order three by their…
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
This paper construct a family of explicit self-similar blowup axisymmetric solutions for the 3D incompressible Euler equations in R^3. Those singular solutions admit infinite energy.
A set of recurrence relations for on-shell two-loop self-energy diagrams with one mass is presented, which allows to reduce the diagrams with arbitrary indices (powers of scalar propagators) to a set of the master integrals. The SHELL2…
In this article, we present a new analytic result for a certain single-mass-scale four-loop vacuum (bubble) integral. We also discuss its systematic $\e$-expansion in $d=4-2\e$ as well as $d=3-2\e$ dimensions, the coefficients of which are…
We consider the scalar sector of a general renormalizable theory and evaluate the effective potential through three loops analytically. We encounter three-loop vacuum bubble diagrams with up to two masses and six lines, which we solve using…
Centered finite volume methods are considered in the context of Numerical Relativity. A specific formulation is presented, in which third-order space accuracy is reached by using a piecewise-linear reconstruction. This formulation can be…
In this paper, we present rotational and self-similar solutions for the compressible Euler equations in R^3 using the separation method. These solutions partly complement Yuen's irrotational and elliptic solutions in R^3 [Commun. Nonlinear…
We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of…
This work presents a detailed account of the Feynman integrals required for the three-loop hadronic vacuum polarization calculation performed in arXiv:2510.12885. We explain how to compute each of the three-loop integrals, and outline the…