The maximal cut of the nonplanar crossed box diagram with all massive internal propagators was long ago shown to encode a hyperelliptic curve genus 3 in momentum space. Surprisingly, Baikov representation, this only gives rise 2. To show that these two representations are agreement, we identify hidden involution symmetry is satisfied by curve, which allows it be algebraically mapped We then argue just first example general mechanism means curves Feynman integrals can drop from <a:math...

Expansion of higher transcendental functions in a small parameter are needed many areas science. For certain classes this can be achieved by algebraic means. These tools based on nested sums and formulated as algorithms suitable for an implementation computer. Examples, such expansions generalized hypergeometric or Appell discussed. As further application, we give the general solution two-loop integral, so-called C-topology, terms multiple sums. In addition, discuss some important properties...

We present the result for finite part of two-loop sunrise integral with unequal masses in four space-time dimensions terms O(ε0)-part and O(ε1)-part around two dimensions. The latter integrals are given elliptic generalisations Clausen Glaisher functions. Interesting aspects occurrence depth objects weights individual terms.

We present the two-loop sunrise integral with arbitrary non-zero masses in two space-time dimensions terms of elliptic dilogarithms. find that structure result is as simple and elegant equal mass case, only arguments dilogarithms are modified. These have a nice geometric interpretation.

The integrand of any multiloop integral is characterized after Feynman parametrization by two polynomials. In this review we summarize the properties these Topics covered in paper include among others: spanning trees and forests, all-minors matrix-tree theorem, recursion relations due to contraction deletion edges, Dodgson's identity matroids.

We present a method to compute the Laurent expansion of two-loop sunrise integral with equal non-zero masses arbitrary order in dimensional regularisation ε. This is done by introducing class functions (generalisations multiple polylogarithms include elliptic case) and showing that all integrations can be carried out within this functions.

Feynman integrals are easily solved if their system of differential equations is in $\varepsilon$-form. In this letter we show by the explicit example kite integral family that an $\varepsilon$-form can even be achieved, do not evaluate to multiple polylogarithms. The obtained a (non-algebraic) change basis for master integrals.

In this paper we show that certain Feynman integrals can be expressed as linear combinations of iterated modular forms to all orders in the dimensional regularisation parameter $\varepsilon$ . We discuss explicitly equal mass sunrise integral and kite integral. For both cases give alphabet letters occurring integrals. present a compact formula, expressing forms.

We show that the Laurent series of two-loop kite integral in D = 4 − 2ε space-time dimensions can be expressed each order expansion terms elliptic generalisations (multiple) polylogarithms. Using differential equations, we present an iterative method to compute any desired order. As example, give first three orders explicitly.

We solve the two-loop sunrise integral with unequal masses systematically to all orders in dimensional regularisation parameter $\varepsilon$. In order do so, we transform system of differential equations for master integrals an $\varepsilon$-form. The depends on three kinematical variables. perform a change variables standard coordinates moduli space ${\mathcal M}_{1,3}$ genus one Riemann surface marked points. This gives us solution as iterated $\overline{\mathcal M}_{1,3}$. On...

Certain Feynman integrals are associated to Calabi-Yau geometries. We demonstrate how these can be computed with the method of differential equations. The four-loop equal-mass banana integral is simplest whose geometry a nontrivial manifold. show that its equation cast into an $ϵ$-factorized form. This allows us obtain solution any desired order in dimensional regularization parameter $ϵ$. generalizes other integrals. Our calculation also shows only minimally more complicated than...

A bstract We describe a systematic approach to cast the differential equation for l -loop equal mass banana integral into an ε -factorised form. With known boundary value at specific point we obtain systematically term of order j in expansion dimensional regularisation parameter any loop . The is based on properties Calabi-Yau operators, and particular self-duality.

I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes particular half-integer values. present algorithms which are suitable for an implementation within symbolic computer algebra system. The method is extension technique nested sums. allow addition evaluation binomial sums, inverse sums and generalizations thereof.

We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed loop Laurent expansion in dimensional regularization parameter ϵ. This is done by transforming system of differential equations this and all its sub-topologies form linear ϵ, where ϵ^{0} part strictly lower triangular. easily solved order an example elliptic multiscale involving several subtopologies. Our methods are applicable similar problems.

We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses.The is obtained by viewing Feynman integral as period of variation mixed Hodge structure, where respect to external momentum squared.The fibre complement an elliptic curve.From fact that first cohomology group this curve two-dimensional we obtain equation.This improvement compared usual way deriving equations: integration-by-parts identities lead only coupled system four...

We study the analytic continuation of Feynman integrals from kite family, expressed in terms elliptic generalisations (multiple) polylogarithms. Expressed this way, are functions two periods an curve. show that all what is required just these periods. present explicit formula for values t∈R. Furthermore, nome q curve satisfies over complete range t inequality |q|≤1, where |q|=1 attained only at singular points t∈{m2,9m2,∞}. This ensures convergence q-series expansion ELi-functions and...

In this paper we exploit factorisation properties of Picard-Fuchs operators to decouple differential equations for multi-scale Feynman integrals. The algorithm reduces the blocks size order irreducible factors operator. As a side product, our method can be used easily convert integrals which evaluate multiple polylogarithms $\varepsilon$-form.

A bstract In this article we give the details on analytic calculation of master integrals for planar double box integral relevant to top-pair production with a closed top loop. We show that these can be computed systematically all order in dimensional regularisation parameter ε . This is done by transforming system differential equations into form linear , where 0 -part strictly lower triangular matrix. Explicit results terms iterated are presented NNLO calculations.

In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express integral in terms polylogarithms, seeks a transformation variables, rationalizes this paper, we give an algorithm for rationalizing The is applicable whenever algebraic hypersurface associated with root has point multiplicity $(d-1)$, where $d$ degree hypersurface. We show that can use iteratively rationalize roots simultaneously. Several examples from high...

Intersection numbers of twisted cocycles arise in mathematics the field algebraic geometry. Quite recently, they appeared physics: define a scalar product on vector space Feynman integrals. With this application, practical and efficient computation intersection becomes topic interest. An existing algorithm for requires intermediate steps introduction extensions (for example, square roots) although final result may be expressed without extensions. In article, I present an improvement...