Differential Equations for Feynman Graph Amplitudes

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to differential equations for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a can...

Integration by parts identities (IBPs) can be used to express large numbers of apparently different d-dimensional Feynman Integrals in terms of a small subset of so-called master integrals (MIs). Using the IBPs one can moreover show that the MIs fulfil linear systems of coupled differential equations in the external invariants. With the increase in number of loops and external legs, one is left in general with an increasing number of MIs and consequently also with an increasi...

An algorithm for the reduction of massive Feynman integrals with any number of loops and external momenta to a minimal set of basic integrals is proposed. The method is based on the new algorithm for evaluating tensor integrals, representation of generalized recurrence relations for a given kind of integrals as a linear system of PDEs and the reduction of this system to a standard form using algorithms proposed by G. Reid. Basic integrals reveal as parametric derivatives of t...

The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in $\epsilon = (4-d)/2$, where $d$ are the space-time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneo...

A survey is given on the present status of analytic calculation methods and the mathematical structures of zero- and single scale Feynman amplitudes which emerge in higher order perturbative calculations in the Standard Model of elementary particles, its extensions and associated model field theories, including effective field theories of different kind.

A new class of identities for Feynman graph amplitudes, dubbed Schouten identities, valid at fixed integer value of the dimension d is proposed. The identities are then used in the case of the two loop sunrise graph with arbitrary masses for recovering the second order differential equation for the scalar amplitude in d=2 dimensions, as well as a chained sets of equations for all the coefficients of the expansions in (d-2). The shift from $d\approx2$ to $d\approx4$ dimensions...

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 used for working out the values at p^2 = 0 and the expansions in p^2 at p^2 =0, in (n-4) at n to 4 limit and in 1/p^2 for large values of p^2 .

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers and provide three different approaches for a direct decomposition of Feynman integrals, which we du...

Integration-by-parts (IBP) identities and differential equations are the primary modern tools for the evaluation of high-order Feynman integrals. They are commonly derived and implemented in the momentum-space representation. We provide a different viewpoint on these important tools by working in Feynman-parameter space, and using its projective geometry. Our work is based upon little-known results pre-dating the modern era of loop calculations: we adapt and generalise these ...

We present a projective framework for the construction of Integration by Parts (IBP) identities and differential equations for Feynman integrals, working in Feynman-parameter space. This framework originates with very early results which emerged long before modern techniques for loop calculations were developed. Adapting and generalising these results to the modern language, we use simple tools of projective geometry to generate sets of IBP identities and differential equatio...