相关论文: Two-loop sunset diagrams with three massive lines
We consider two loop sunset diagrams with two mass scales m and M at the threshold and pseudotreshold that cannot be treated by earlier published formula. The complete reduction to master integrals is given. The master integrals are…
We derive exact, convergent representations of multiloop sunset Feynman integrals in two dimensions for arbitrary mass configurations and all loop orders valid for large Euclidean momentum. The integrals are expressed as sums of symmetric…
We present the two-loop sunrise integral with arbitrary non-zero masses in two space-time dimensions in terms of elliptic dilogarithms. We find that the structure of the result is as simple and elegant as in the equal mass case, only the…
In this paper we consider two-loop two-, three- and four-point diagrams with elliptic structure in the case of two different masses $m$ and $M$. The latter diagrams generally arise within NRQCD matching procedures and are relevant for…
The 4-loop sunrise graph with two massless lines, two lines of equal mass M and a line of mass m, for external invariant timelike and equal to m^2 is considered. We write differential equations in x=m/M for the Master Integrals of the…
We show how to compute the two-loop sunset integrals at finite volume, for non-degenerate masses and non-zero momentum. We present results for all integrals that appear in the Chiral Perturbation Therory ($\chi$PT) calculation of the…
The scalar two-loop self-energy master diagram is studied in the case of arbitrary masses. Analytical results in terms of Lauricella- and Appell-functions are presented for the imaginary part. By using the dispersion relation a…
In these notes the relativistic $n$-body phase-phase is calculated iteratively in $2+1$ space-time dimensions for all $n$. The obtained result shows a simple power-law behavior $\alpha_n (\mu-M)^{n-2}/\mu$ with a dependence only on the…
We present the results for two-loop massive kite master integrals with elliptics in terms of iterated integrals with algebraic kernels. The key ingredients are new integral representations for sunset subgraphs in $d=4-2\epsilon$ and…
We compute the (three) master integrals for the crossed ladder diagram with two exchanged quanta of equal mass. The differential equations obeyed by the master integrals are used to generate power series expansions centered around all the…
We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic…
The two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order…
We determine the numerical values of scalar multi-loop two-vertex Feynman diagrams, the generalized sunset diagrams, by integrating all but the longitudinal momenta analytically. For the longitudinal momenta we introduce one collective…
The massless sunrise diagram with an arbitrary number of loops is calculated in a simple but formal manner. The result is then verified by rigorous mathematical treatment. Pitfalls in the calculation with distributions are highlighted and…
The master differential equations in the external square momentum p^2 for the master integrals of the two-loop sunrise graph, in n-continuous dimensions and for arbitrary values of the internal masses, are derived. The equations are then…
In this talk, we discuss our recent computation of the two-loop sunrise integral with arbitrary non-zero particle masses. In two space-time dimensions, we arrive at a result in terms of elliptic dilogarithms. Near four space-time…
The techniques of integration by parts and differential reduction differ in the counting of master integrals. This is illustrated using as an example the two-loop sunset diagram with on-shell kinematics. A new algebraic relation between the…
We present the result for the finite part of the two-loop sunrise integral with unequal masses in four space-time dimensions in terms of the ${\mathcal O}(\varepsilon^0)$-part and the ${\mathcal O}(\varepsilon^1)$-part of the sunrise…
Sunset integrals are among the simplest of two-loop integrals that appear in perturbative quantum field theories and possess up to four distinct mass scales. By means of integration by parts identities, they can be written in terms of four…
Small momentum expansion of the "sunset" diagram with three different masses is obtained. Coefficients at powers of $p^2$ are evaluated explicitly in terms of dilogarithms and elementary functions. Also some power expansions of "sunset"…