Non-planar Feynman diagrams and Mellin-Barnes representations with AMBRE 3.0

We introduce the Mellin-Barnes representation of general Feynman integrals and discuss their evaluation. The Mathematica package AMBRE has been recently extended in order to cover consistently non-planar Feynman integrals with two loops. Prospects for the near future are outlined. This write-up is an introduction to new results which have also been presented elsewhere.


Introduction
The evaluation of Feynman integrals is a central numerical problem of perturbative quantum field theory and is not solved in generality beyond the one-loop case. One has to consider arbitrary L-loop integrals G(X) with loop momenta k l , with E external legs with momenta p e , and with N internal lines with masses m i and propagators 1/D i , c l i k l + E e=1 d e i p e ] − m 2 i , and some tensors X(k 1 , . . . , k L ) in the loop momenta. For several reasons, one is interested in analytical solutions either: • Analytical results are well suited for the exact cancellation of certain intermediate terms, arising either from the renormalization procedure, from regularization, or just from the organization of the calculation. • Analytical results may stabilize the numerics.
• With analytical results, analytical continuations of representations of Feynman integrals into regions of physical interest may be performed. Typically, it is easier to determine a Feynman integral in the Euclidean region, but often it is needed in the Minkowskian.
• Alternatively, a purely numerical evaluation of the multi-dimensional Mellin-Barnes integral may be envisaged. Here it is wishful to cover not only Euclidean cases, but also the Minkowskian kinematics; this has not been studied so far.
Of course, a solution of the general case is not to be expected. The limitations of the method have several reasons; we mention here: • The number of loops; • The number of different scales due to internal masses and the kinematics; • The number of external legs; • A planar or non-planar topology.
A Feynman parameter integral with N internal legs has, essentially, a dimensionality N − 1. For the corresponding MB-integral, the dimensionality will be different, and for complex problems it often will come out much higher.
Here it becomes operational to apply the Cheng-Wu theorem [53,54] which states that (2) holds also with a modified delta function δ(1 − i∈Ω x i ) where Ω is an arbitrary subset of the lines 1, . . . , N , when the integration over the rest of the variables, i.e. for i / ∈ Ω, is extended to the integration from 0 to ∞. With AMBRE 3.0, non-planar Feynman integrals with two loops may be efficiently represented by a direct approach with use of the Cheng-Wu theorem. Planar integrals are treated by the loop-by-loop ansatz with the earlier AMBRE versions. For the example of the massless non-planar double box mentioned, AMBRE 3.0 derives without manual interaction the 4-dimensional Mellin-Barnes representation of [24]. It might look more promising to apply here instead the loop-by-loop approach, but experience shows that the dimensionality becomes, without further interventions, higher. At the other hand, for the massive non-planar double box, the loop-by-loop approach gives the MB-presentation derived first in [25]. We mention here shortly that 3-loop integrals may be treated in a hybrid way by a loop-by-loop approach, subsequently using AMBRE 1.2 and AMBRE 3.0 [50]; see the example files MB hybrid 3loopNP massless.nb, MB hybrid 3loopNP massless.m, out MB hybrid 3loopNP massless at the webpage http://prac.us.edu.pl/∼gluza/ambre/.
Finally we would like to mention that the mathematica package AMBRE 3.0 is released for public use at the webpage http://prac.us.edu.pl/∼gluza/ambre/. There is no source file made publicly available so far, but it may be made available on request. We consider this to be appropriate; for closer information we refer to the webpages http://fh.desy.de/projekte/-gfitter01/Gfitter01.htm (March 2013) and http://zfitter-gfitter.desy.de/ (April 2014). For a collection of experts' views on proper distribution of scientific software in basic research, we would like to refer to [55,56], a summary of a round table discussion at ACAT 2014.

Summary
We scetched essential features of the Mellin-Barnes approach to Feynman integrals and its implementation in AMBRE. In the new version AMBRE 3.0 the Cheng-Wu theorem is implemented and the replacement of the Symanzik polynomials by MB-integrals is performed globally. The alternative loop-by-loop approach is implemented in AMBRE 2.0 or older versions. Which of the approaches is more appropriate for a given problem has to be investigated.
A treatment of non-planar three-loop integrals deserves the combined application of AMBRE 1.2 and of AMBRE 3.0 in a hybrid approach. This will be improved in the nearest future. The analytical summation of series of MB-residues is under study. Concerning the alternative to analytical summation, namely a numerical evaluation of the MB-integrals, we are restricted with AMBRE to the Euclidean kinematics so far. The Minkowskian numerics is on our to-do list.