Cheat sheets for many-body perturbation theory

This repository summarizes the analytic expressions for many-body perturbation theory (MBPT) through sixth order (third order) at the normal-ordered two-body (three-body) level. LaTeX source files and prebuilt PDF files are provided (see `pdf` and `tex` directories, respectively).

We hope that these documents will serve as useful cheat sheets for learning, education, and research based on MBPT (i.e., intended as helpful quick references), whether in nuclear physics or related scientific fields.

For applications to the equation of state of infinite nuclear matter with chiral two- and three-nucleon interactions, e.g., see

• C. Drischler, K. S. McElvain, and P. Arthuis, arXiv:2603.24532
• C. Drischler, J. W. Holt, and C. Wellenhofer, Annu. Rev. Nucl. Part. Sci. 71, 403 (2021), arXiv:2101.01709.
• C. Drischler, K. Hebeler, and A. Schwenk, Phys. Rev. Lett. 122, 042501 (2019), arXiv:1710.08220.

The analytic expressions associated with the following MBPT orders and levels are contained in this repository:

MBPT order	MBPT level	Number of diagrams1	PDF
0 (Hartree Fock)	1,2,3,4	4	link
2	2	1	link
3	2	3	link
4	2	39	link
5	2	840	link
6	2	27300	link
2	3	1	link
3	3	14	link

Building the PDF files

To (re)build all PDF files in the directory pdf from the LaTeX files, simply run in the root folder of this repository.

make

Make sure that latexmk, which comes with many LaTeX distributions, is installed. This may take a while, especially due to the many diagrams at sixth order in MBPT. Each file is built several times during the process to update the table of contents and internal hyperlinks.

The LaTeX interpreter lualatex is used by default. To use pdflatex instead, simply run

make TEX=pdflatex

However, the sixth-order expressions can only be built using lualatex since the file exceeds internal limits of pdflatex.

To remove the auxiliary files generated by LaTeX, run

make clean

References

The fifth- and sixth-order expressions were generated using automated diagram generation as described in the following publication, and kindly provided by Pierre Arthuis.

@article{Arthuis:2018yoo,
    author = "Arthuis, P. and Duguet, T. and Tichai, A. and Lasseri, R.-D. and Ebran, J.-P.",
    title = "{ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory}",
    eprint = "1809.01187",
    archivePrefix = "arXiv",
    primaryClass = "nucl-th",
    doi = "10.1016/j.cpc.2018.11.023",
    journal = "Comput. Phys. Commun.",
    volume = "240",
    pages = "202--227",
    year = "2019"
}

The source code for the automated diagram generator is publicly available:

@misc{adg,
    title = {\texttt{ADG} -- automated diagram generator},
    author = {Arthuis, Pierre},
    year = "2019",
    url = {https://adg.readthedocs.io/en/master/project.html},
    note = {Python package available for download from GitHub,
      \href{https://adg.readthedocs.io/en/master/project.html}{https://adg.readthedocs.io/en/master/project.html}}
}

The diagrams for the residual three-body terms at third order are given in Figure 1 in:

@misc{Hu:2018dza,
    author = "Hu, B.S. and Li, T. and Xu, F.R.",
    title = "{Ab initio Rayleigh-Schrödinger perturbation calculation including three-body force}",
    eprint = "1810.08804",
    archivePrefix = "arXiv",
    primaryClass = "nucl-th",
    month = "10",
    year = "2018"
}

The analytic expressions have been checked with the authors. Many thanks!

Citing this work

This repository is part of our manuscript:

@article{Drischler:2026vdm,
    author = "Drischler, C. and McElvain, K. S. and Arthuis, P.",
    title = "{Many-body perturbation theory for the nuclear equation of state up to fifth order}",
    eprint = "2603.24532",
    archivePrefix = "arXiv",
    primaryClass = "nucl-th",
    month = "3",
    year = "2026"
}

To cite this repository, please use:

@software{Drischler_Cheat_sheets,
author = {Drischler, Christian and McElvain, Kenneth S. and Arthuis, Pierre},
month = {4},
title = {Cheat sheets for many-body perturbation theory},
note = {available on {GitHub}: \url{https://github.com/cdrischler/mbpt-cheat-sheets}},
url = {https://github.com/cdrischler/mbpt-cheat-sheets},
version = {1.0.0},
year = {2026}
}

Contact

Christian Drischler
Department of Physics and Astronomy
Clippinger Laboratories 265
139 University Terrace
1 Ohio University Drive
Athens, OH 45701-2979
Email: drischler@ohio.edu

Pierre Arthuis
Laboratoire de Physique des 2 infinis Irène Joliot-Curie – IJCLab
Bâtiment 100, Pièce A206
15 rue Georges Clémenceau
91405 Orsay cedex
Email: pierre.arthuis@ijclab.in2p3.fr

Acknowledgments

This material is based upon work supported by the NSF under award PHY-2339043 and the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under the FRIB Theory Alliance award DE-SC0013617.

Footnotes

1. The number of diagrams at the two-body level is known as the OEIS integer sequence A064732 "Number of labeled Hugenholtz diagrams with n nodes." ↩