Contacts: alex.radionov89@gmail.com fyodor.tkachov@gmail.com
A general context for this project is systematic large-scale computations of two-loop integrals -- anything beyond two loops is outside our scope.
A specific aim is to find practical ways to construct D-operators for the generalized IBP reduction method (also called the BST method, see sec. 3.3.4 of a recent review by Gudrun Heinrich, https://inspirehep.net/literature/1814379).
The theology of the project was summarized in the talk at the Bogolyubov-2019 conference, see below.
The Bernstein-Sato equation proves to be full of surprises that keep popping up. Therefore we feel it to be counterproductive to focus on a specific class of diagrams etc. The examples presented below were not produced purposefully or systematically, they just came about from playing with the software at a particular stage of its development, and are presented as is.
2020-05-30
A (still another) surprising finding was made while playing with alternative
representations of numerical coefficients. It turns out there is a heuristic way
to guess an overall structure of the (partial) D-operator in a specific
situation. Such knowledge speeds up the calculation of the D-operator very
significantly.
The document Hat1234.pdf contains a Mathematica
verification of the first (partial) D0-operator thus obtained. It is for the
two-loop dunce's cap diagram that has three independent Feynman-Chisholm
integration parameters. The four mass-squared parameters were set to 1, 2, 3,
and 4.
The computation time was not recorded as the result was unexpected.
Note that the corresponding b-function (the innermost round brackets in the line
In[88]) factorizes into three simple factors:
The overall m0 was expected (see the text
"Partial ..." below).
We have no explanation for the other factorization.
2020-05-17
Partial D-operators for the generalized IBP reduction
https://arxiv.org/abs/1912.04857
2019-12-10
Contribution to the Bogolyobov-2019 conference proceedings, authors' English
version:
https://arxiv.org/abs/1912.04857
Supplements:
First examples of the D-operators for the simplest two-loop diagram.
The diagram and the two polynomials --
pdf.
Two partial D-operators for all masses equal to 1, k^2=3 --
txt.
Two partial D-operators with k-dependence restored, all masses equal to 1 --
pdf.
For completeness: a Mathematica check of the GRACE-2012 result -- pdf.
2019-09-12
Talk at the Bogolyubov-2019 Conference,
JINR, Dubna.
A.A. Radionov and F.V. Tkachov. Breaking the 2-loop barrier for the generalized
IBP reduction method.
Presentation: pdf (16 pp, 0.5M): see
above for links to examples in separate files.
Video (in Russian at the organizers' and audience's desire):
https://youtu.be/vXcGHjxcdJM?t=28060
The initial publication (1996):
F.V. Tkachov. Algebraic algorithms for multiloop calculations. The First 15
years. What's next?
http://inspirehep.net/record/423614
Mathematical background:
Bernstein's 1972 paper:
по-русски,
English
translation.
A beginner's textbook:
S. C. Coutinho. A Primer of Algebraic D-Modules. Cambridge, 1995.
