# blade

**Repository Path**: multiloop-pku/blade

## Basic Information

- **Project Name**: blade
- **Description**: Block-triangular form improved Feynman integral decomposition .
- **Primary Language**: Unknown
- **License**: MIT
- **Default Branch**: main
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 0
- **Forks**: 0
- **Created**: 2025-01-24
- **Last Updated**: 2026-05-10

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

# Blade

**Blade** is a **BL**ock-tri**A**ngular form improved Feynman integral **DE**composition package that uses a search algorithm to construct a system of simple relations among Feynman integrals, using input from integration-by-parts (IBP) decomposition at specific numerical phase space points over a finite field. The system usually has orders of magnitude fewer equations than traditional IBP method and exhibits a block triangular form, which leads to fast numerical evaluation and reduced resource consumption.

The search algorithm is based on the following publications:
- Xiao Liu and Yan-Qing Ma. Determining arbitrary Feynman integrals by vacuum integrals. [[Phys.Rev.D 99 (2019) 071501](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.99.071501)]([arXiv: 1801.10523](https://arxiv.org/abs/1801.10523))
- Xin Guan, Xiao Liu and Yan-Qing Ma. Complete reduction of integrals in two-loop five-light-parton scattering amplitudes. [[Chin.Phys.C 44 (2020) 9, 093106](https://iopscience.iop.org/article/10.1088/1674-1137/44/9/093106)]([arXiv: 1912.09294](https://arxiv.org/abs/1912.09294))
- Xin Guan, Xiao Liu, Yan-Qing Ma and Wen-Hao Wu. Blade: A package for block-triangular form improved Feynman integrals
decomposition. [[Comput.Phys.Commun. 310 (2025) 109538](https://www.sciencedirect.com/science/article/pii/S0010465525000414?via%3Dihub)]([arXiv: 2405.14621](https://arxiv.org/abs/2405.14621))

## Installation

If you use Ubuntu system and have root privilege, you can install Blade as well as its dependencies (except `Mathematica`) fully automatically. To this end, download the `auto_install` file (such as using the command `wget https://gitee.com/multiloop-pku/blade/raw/main/auto_install`) and then run
```
chmod +x auto_install
./auto_install
```
You might not be running the command as root, such as using `sudo ./auto_install`, which could result in the file being owned by root instead of the current user.

If you use macOS, we also provide an automatic installation script based on `Homebrew`. The script installs Blade together with its dependencies (except `Mathematica`). Download the script (for example using `wget https://gitee.com/multiloop-pku/blade/raw/main/auto_install_macos`) and then run
```
chmod +x auto_install_macos
./auto_install_macos
```
For users who do not have root privileges or use different operation system, we provide a sample installation document `sample_install.md` that users can follow. 

### Dependencies

We use FiniteFlow ([arXiv:1905.08019](https://arxiv.org/abs/1905.08019)) as a sparse solver and finite-field reconstructor. The following packages should be installed in order.

- [Mathematica](https://www.wolfram.com/)

- [CMake](https://cmake.org/)

- [GMP](https://gmplib.org/)

- [MPFR](https://www.mpfr.org/)

- [FLINT](http://www.flintlib.org/) 

- [FiniteFlow](https://github.com/peraro/finiteflow/)

### Optional Dependencies

- [Ratracer](https://github.com/magv/ratracer)

    We support using Ratracer ([arXiv:2211.03572](https://arxiv.org/abs/2211.03572)) for finite field reconstruction.
    Ratracer itself relies on the [FireFly](https://gitlab.com/firefly-library/firefly) library. 
    
    To use Ratracer, you should reset the path listed in BLAddOns/RatRacer/install.m.


### Installation of Blade

To set up the paths for dependencies, you should edit the following information in the `install.in.txt` file:
```
DFFLOWMLINK_DIR="/path/to/finiteflow-master"
DCMAKE_PREFIX_PATH="/dependencies/installation/path/prefix"
```
The `/path/to/finiteflow-master` directory should contain the dynamic-link library `libfflow.so`(or `libfflow.dylib`), the mathlink file `fflowmlink.so`(or `fflowmlink.dylib`), and the Mathematica file for FiniteFlow `mathlink/FiniteFlow.m`.
\
The `/dependencies/installation/path/prefix` refers to the installation paths for `GMP`, `MPFR` and `FLINT`, separated by semicolons. You can omit any of the directories if the corresponding dependencies were installed in the default installation path (e.g., /usr/local).

Then to install the binaries and generate the necessary files, use the following command:
```
cmake .
make install
```

To remove the files generated during the `cmake` and `make install` processes, use the following command:
```
./uninstall
```

To update Blade to the latest version, use the following command:
```
./update
```

Additional instructions for MacOS users can be found in the `install.in.txt` file.


## Usage

Several examples are provided in `example` folder, in the order of complexity. You are encouraged to run examples that name begin with `1_` to get familiar with the package. 

- 1_automatic - introduction to automatic reduction of Feynman loop integrals;
- 1_preferred_masters - introduction to automatic reduction with user-defined master integrals;
- 1_userdefined_target - introduction to automatic reduction of user defined target integrals;
- 1_differential_equations - introduction to automatic construction of differential equations;
- 1_prescription - introduction to automatic reduction of Feynman integrals with cut propagators;
- 1_complex_number - introduction to reduction with complex replacement rules;
- 1_improve_masters - introduction to d-factorized basis and physical basis;
- 1_spanning_reduce - introduction to the spanning-sector reduction algorithm;
- 1_blade_vs_plain_ibp - comparision between the block-triangular form improved reduction and the traditional IBP reduction;
- 1_generalized_integrand - introduction to automatic reduction of Feynman integrals with additional integrand;
- 2_reducer_options - introduction to a few options;
- 2_reconstructor_option - introduction to the interface for Ratracer;
- 2_block_triangular_form - introduction to the block-triangular form;
- 3_manual_search_options - introduction to the manual reduction with adaptive-search algorithm;
- 4_paper_higgsjet - a three-loop four-point diagram with one massive external line;
- 4_paper_fsf3 - a four-loop two-point diagram with one massive internal particle;
- 4_paper_topo5 - a two-loop four-point diagram with two massive internal particle;
- 4_paper_dpmass - a two-loop five-point diagram with one massive external line.

## Support

If you have any questions or advices, please do not hesitate to contact us: guanxin0507@pku.edu.cn, xiao.liu@physics.ox.ac.uk, yqma@pku.edu.cn, wuwenhao21@mails.ucas.ac.cn