# 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. doi: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. doi: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

DOI: 10.1103/PhysRevLett.123.151601

DOI: 10.1103/PhysRevLett.123.151601

Other availability

Cited by: 14 works

*Citation information provided by*

Web of Science

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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Works referenced in this record:

##
Minimal anomaly-free chiral fermion sets and gauge coupling unification

journal, December 2014

- Cebola, Luís M.; Emmanuel-Costa, D.; Felipe, R. González
- Physical Review D, Vol. 90, Issue 12

##
An anomaly-free atlas: charting the space of flavour-dependent gauged U(1) extensions of the Standard Model

journal, February 2019

- Allanach, B. C.; Davighi, Joe; Melville, Scott
- Journal of High Energy Physics, Vol. 2019, Issue 2

##
Number-theory dark matter

journal, May 2011

- Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T.
- Physics Letters B, Vol. 699, Issue 5

##
Imperfect mirror copies of the standard model

journal, August 2016

- Berryman, Jeffrey M.; de Gouvêa, André; Hernández, Daniel
- Physical Review D, Vol. 94, Issue 3

##
Axial vector Z ′ and anomaly cancellation

journal, May 2017

- Ismail, Ahmed; Keung, Wai-Yee; Tsao, Kuo-Hsing
- Nuclear Physics B, Vol. 918

##
Effect of Anomalies on Quasi-Renormalizable Theories

journal, July 1972

- Gross, David J.; Jackiw, R.
- Physical Review D, Vol. 6, Issue 2

##
Simple model for (3+2) neutrino oscillations

journal, July 2004

- Babu, K. S.; Seidl, Gerhart
- Physics Letters B, Vol. 591, Issue 1-2

##
Anomalous Ward Identities in Spinor Field Theories

journal, August 1969

- Bardeen, William A.
- Physical Review, Vol. 184, Issue 5

##
Dark discrete gauge symmetries

journal, February 2011

- Batell, Brian
- Physical Review D, Vol. 83, Issue 3

##
Fermion mass hierarchy and proton stability from nonanomalous $U(1{)}_{F}$ in supersymmetric $SU(5)$

journal, July 2008

- Chen, Mu-Chun; Jones, D. R. Timothy; Rajaraman, Arvind
- Physical Review D, Vol. 78, Issue 1

##
Gauge Theories Without Anomalies

journal, July 1972

- Georgi, Howard; Glashow, Sheldon L.
- Physical Review D, Vol. 6, Issue 2

##
Surprises from complete vector portal theories: New insights into the dark sector and its interplay with Higgs physics

journal, November 2017

- Cui, Yanou; D’Eramo, Francesco
- Physical Review D, Vol. 96, Issue 9

##
Anomaly-free sets of fermions

journal, August 2006

- Batra, Puneet; Dobrescu, Bogdan A.; Spivak, David
- Journal of Mathematical Physics, Vol. 47, Issue 8

##
Axial-Vector Vertex in Spinor Electrodynamics

journal, January 1969

- Adler, Stephen L.
- Physical Review, Vol. 177, Issue 5

##
New chiral fermions, a new gauge interaction, Dirac neutrinos, and dark matter

journal, October 2015

- de Gouvêa, André; Hernández, Daniel
- Journal of High Energy Physics, Vol. 2015, Issue 10

##
Gravitation, gauge theories and differential geometry

journal, December 1980

- Eguchi, Tohru; Gilkey, Peter B.; Hanson, Andrew J.
- Physics Reports, Vol. 66, Issue 6

##
Models of neutrino mass with a low cutoff scale

journal, June 2005

- Davoudiasl, Hooman; Kitano, Ryuichiro; Kribs, Graham D.
- Physical Review D, Vol. 71, Issue 11

##
An anomaly-free version of Weinberg's model

journal, April 1972

- Bouchiat, C.; Iliopoulos, J.; Meyer, Ph.
- Physics Letters B, Vol. 38, Issue 7

Works referencing / citing this record:

##
Hypercharge quantisation and Fermat's last theorem

journal, January 2020

- Lohitsiri, Nakarin; Tong, David
- SciPost Physics, Vol. 8, Issue 1

##
Anomaly free Froggatt-Nielsen models of flavor

journal, October 2019

- Smolkovič, Aleks; Tammaro, Michele; Zupan, Jure
- Journal of High Energy Physics, Vol. 2019, Issue 10

##
Notes on anomalies, elliptic curves and the BS-D conjecture

journal, December 2019

- Lü, Yongchao; Minahan, Joseph A.
- Journal of Physics A: Mathematical and Theoretical, Vol. 53, Issue 2

##
Classification of anomaly-free 2HDMs with a gauged $U(1{)}^{\prime}$ symmetry

journal, December 2019

- Nottensteiner, Franz; Ordell, Astrid; Pasechnik, Roman
- Physical Review D, Vol. 100, Issue 11

##
Gauging the accidental symmetries of the standard model, and implications for the flavor anomalies

journal, January 2020

- Altmannshofer, Wolfgang; Davighi, Joe; Nardecchia, Marco
- Physical Review D, Vol. 101, Issue 1

##
Chiral Froggatt-Nielsen models, gauge anomalies and flavourful axions

text, January 2020

- Bonnefoy, Quentin Rene Christian; Dudas, E.; Pokorski, S.
- Deutsches Elektronen-Synchrotron, DESY, Hamburg

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