# General Solution to the U(1) Anomaly Equations

## Abstract

The anomaly cancellation equations for the $U(1)$ gauge group can be written as a cubic equation in $n-1$ integer variables, where $$n$$ is the number of Weyl fermions carrying the $U(1)$ charge. We solve this Diophantine cubic equation by providing a parametrization of the charges in terms of $n-2$ integers, and prove that this is the most general solution.

- Authors:

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP)

- OSTI Identifier:
- 1571315

- Alternate Identifier(s):
- OSTI ID: 1568881

- Report Number(s):
- arXiv:1905.13729; FERMILAB-PUB-19-252-T

Journal ID: ISSN 0031-9007; PRLTAO; 151601

- Grant/Contract Number:
- AC02-07CH11359

- Resource Type:
- Published Article

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Name: Physical Review Letters Journal Volume: 123 Journal Issue: 15; Journal ID: ISSN 0031-9007

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

## Citation Formats

```
Costa, Davi B., Dobrescu, Bogdan A., and Fox, Patrick J. General Solution to the U(1) Anomaly Equations. United States: N. p., 2019.
Web. doi:10.1103/PhysRevLett.123.151601.
```

```
Costa, Davi B., Dobrescu, Bogdan A., & Fox, Patrick J. General Solution to the U(1) Anomaly Equations. United States. https://doi.org/10.1103/PhysRevLett.123.151601
```

```
Costa, Davi B., Dobrescu, Bogdan A., and Fox, Patrick J. Wed .
"General Solution to the U(1) Anomaly Equations". United States. https://doi.org/10.1103/PhysRevLett.123.151601.
```

```
@article{osti_1571315,
```

title = {General Solution to the U(1) Anomaly Equations},

author = {Costa, Davi B. and Dobrescu, Bogdan A. and Fox, Patrick J.},

abstractNote = {The anomaly cancellation equations for the $U(1)$ gauge group can be written as a cubic equation in $n-1$ integer variables, where $n$ is the number of Weyl fermions carrying the $U(1)$ charge. We solve this Diophantine cubic equation by providing a parametrization of the charges in terms of $n-2$ integers, and prove that this is the most general solution.},

doi = {10.1103/PhysRevLett.123.151601},

journal = {Physical Review Letters},

number = 15,

volume = 123,

place = {United States},

year = {2019},

month = {10}

}

Free Publicly Available Full Text

Publisher's Version of Record

https://doi.org/10.1103/PhysRevLett.123.151601

https://doi.org/10.1103/PhysRevLett.123.151601

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Cited by: 33 works

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Web of Science

#### Figures / Tables:

TABLE I: Primitive solutions to the anomaly equations for 6 fermions in canonical form for $z$

_{1}≤ 12. A choice of the $k$_{1}, $k$_{2},*ℓ*_{1},*ℓ*_{2}parameters and the greatest common divisor that generate the primitive solution from the set (17) are shown.
All figures and tables
(1 total)

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*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*