# Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral

## Abstract

We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure mathematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral in question. These kernels allow us to express all iterated integrals on an elliptic curve in terms of them. The flexibility of our approach leads us to expect that it will be applicable to a large variety of integrals in high-energy physics.

- Authors:

- Humboldt Univ. of Berlin (Germany). Inst. of Mathematics. Inst. of Physics
- European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.; Univ. of Louvain (UCL), Louvain-la-Neuve (Belgium). Center for Cosmology, Particle Physics and Phenomenology (CP3)
- SLAC National Accelerator Lab., Menlo Park, CA (United States)
- European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States); European Organization for Nuclear Research (CERN), Geneva (Switzerland); Humboldt Univ. of Berlin (Germany)

- Sponsoring Org.:
- USDOE; European Research Council (ERC); German Research Foundation (DFG)

- OSTI Identifier:
- 1457782

- Grant/Contract Number:
- AC02-76SF00515; 637019

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 11; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; electroweak radiative corrections; perturbative QCD; quantum electrodynamics

### Citation Formats

```
Broedel, Johannes, Duhr, Claude, Dulat, Falko, and Tancredi, Lorenzo.
```*Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral*. United States: N. p., 2018.
Web. doi:10.1103/physrevd.97.116009.

```
Broedel, Johannes, Duhr, Claude, Dulat, Falko, & Tancredi, Lorenzo.
```*Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral*. United States. doi:10.1103/physrevd.97.116009.

```
Broedel, Johannes, Duhr, Claude, Dulat, Falko, and Tancredi, Lorenzo. Tue .
"Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral". United States.
doi:10.1103/physrevd.97.116009. https://www.osti.gov/servlets/purl/1457782.
```

```
@article{osti_1457782,
```

title = {Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral},

author = {Broedel, Johannes and Duhr, Claude and Dulat, Falko and Tancredi, Lorenzo},

abstractNote = {We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure mathematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral in question. These kernels allow us to express all iterated integrals on an elliptic curve in terms of them. The flexibility of our approach leads us to expect that it will be applicable to a large variety of integrals in high-energy physics.},

doi = {10.1103/physrevd.97.116009},

journal = {Physical Review D},

number = 11,

volume = 97,

place = {United States},

year = {Tue Jun 12 00:00:00 EDT 2018},

month = {Tue Jun 12 00:00:00 EDT 2018}

}