Computing Master Feynman Integrals with Semi-Definite Programming

Introduction

This repository serves as a simple implementation for the paper Feynman Integrals from Positivity Constraints. See intro.pdf for a brief formal introduction.

Getting Started

If all prerequisites are already satisfied, simply run

make

followed by

./master <config_file.yaml>

to launch master integral evaluation. Please check examples/ subdirectory for configure file format.

The dependencies gsl and sdpa are optional. Without gsl, the Euclidean region check is disabled. To compile without gsl, run

make NO_GSL=true

Without sdpa, the program cannot actually evaluate master integrals. Instead, it outputs a semi-definite program in plain text. The output format is compatible with sdpa, so that one may process it later with an external sdpa solver to get the values of master integrals. To compile without sdpa, run

make NO_SDPA_LIB=true

Prerequisites

The "Instructions" sections are tested on Ubuntu 22.04 LTS.

• yaml-cpp (https://github.com/jbeder/yaml-cpp)

• Fermat 6.5 (http://home.bway.net/lewis/)

  • Instructions: Download binary version of Fermat 6.5 here and extract. Then set the environment variable FERMATPATH to the path to Fermat executable fer64, namely

    export FERMATPATH="/path/to/ferl6/fer64"

• GiNaC (https://www.ginac.de/), which itself depends on CLN (https://www.ginac.de/CLN/)

  • Instructions: To build CLN and GiNaC from source, clone repositories

    git clone git://www.ginac.de/cln.git
    git clone git://www.ginac.de/ginac.git

    and check the respective INSTALL files for installation instructions.

• Kira 2.3 (https://gitlab.com/kira-pyred/kira)

  • Instructions: Clone Kira repository

    git clone https://gitlab.com/kira-pyred/kira.git

    and check README.rst for installation instructions.

• GSL (GNU Scientific Library) 2.8 (optional) (https://www.gnu.org/software/gsl/)

• SDPA 7.3.15 (optional) (https://sdpa.sourceforge.net/)

  • Instructions: The binary version of SDPA can be installed by apt-get,

    # To install SDPA,
    sudo apt-get install sdpa
    # To install SDPA callable libraries and examples,
    sudo apt-get install libsdpa-dev

Reproducing Results

To reproduce the one-loop bubble integral results in Feynman Integrals from Positivity Constraints, namely equations (2.79), (2.103) and (2.107), compile and run

./master examples/bubble.yaml

To reproduce the three-loop unequal-mass banana integral results, namely equations (3.35)~(3.37), compile and run

./master examples/banana.yaml