On the deformation of abelian integrals

We consider the deformation of abelian integrals which arose from the study of SG form factors. Besides the known properties they are shown to satisfy Riemann bilinear identity. The deformation of intersection number of cycles on hyperelliptic curve is introduced.Comment: 8 pages, AMSTE

Geometric approach to asymptotic expansion of Feynman integrals

We present an algorithm that reveals relevant contributions in non-threshold-type asymptotic expansion of Feynman integrals about a small parameter. It is shown that the problem reduces to finding a convex hull of a set of points in a multidimensional vector space.Comment: 6 pages, 2 figure

Evaluating single-scale and/or non-planar diagrams by differential equations

We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, $p_2^2\neq 0$. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with $\epsilon=(4-D)/2$. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small $p_2^2$ to our results at $p_2^2\neq 0$ and to identify various terms in these solutions according to expansion by regions. The second family consists of four-point non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called $K_4$ graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in $\epsilon$ up to weight six.Comment: 27 pages, 2 figure