The Mathematics of Linear Relations between Feynman Integrals

18-22 March 2019
Mainz Institute for Theoretical Physics, Johannes Gutenberg University
Europe/Berlin timezone

Computation of holonomic systems for Feynman amplitudes associated with some simple diagrams

19 Mar 2019, 09:30
1h
02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

02.430

Mainz Institute for Theoretical Physics, Johannes Gutenberg University

Staudingerweg 9 / 2nd floor, 55128 Mainz

Speaker

Prof. Toshinori Oaku (Tokyo Woman's Christian University)

Description

Holonomic systems are a class of systems of linear (partial or ordinary) differential equations. One of the most fundamental properties of a holonomic system is that its solution space is finite-dimensional.

A Feynman amplitude is the integral of a rational function, or more generally, the product of complex powers of polynomials.
Hence we can, in principle, apply the following two facts in (computational) $D$-module theory:

1.
For (multivariate) polynomials $f_1,\dots,f_d$ and complex numbers $\lambda_1,\dots,\lambda_d$, the multi-valued analytic function $f_1^{\lambda_1}\cdots f_d^{\lambda_d}$ satisfies a holonomic system, which can be computed algorithmically.

2.
If a function satisfies a holonomic system, its integral with respect to some of its variables also satisfies a holonomic system, which can be computed algorithmically.

In the integration, it would also be natural to ragard the integrand as a local cohomology class associated with $f_1,\dots,f_d$, which roughly corresponds to the 'residue' of
$f_1^{\lambda_1}\cdots f_d^{\lambda_d}$ at $\lambda_1 = \cdots = \lambda_d = -1$ (at least in positive mass case for external diagrams).
There are also algorithms for computing a holonomic system for such a cohomology class.
However, actual computation, especially of integration, is hard in general because of the complexity.
I shall present some worked out examples together with an interpretation based on microlocal analysis.

Primary author

Prof. Toshinori Oaku (Tokyo Woman's Christian University)