# Lectures on differential equations for Feynman integrals

## Abstract

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space–time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE.

- Authors:

- Institute for Advanced Study, Princeton, NJ (United States)

- Publication Date:

- Research Org.:
- Institute for Advanced Study, Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Advanced Research Projects Agency - Energy (ARPA-E)

- OSTI Identifier:
- 1341773

- Alternate Identifier(s):
- OSTI ID: 1454731

- Grant/Contract Number:
- SC0009988

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- Journal of Physics. A, Mathematical and Theoretical

- Additional Journal Information:
- Journal Volume: 48; Journal Issue: 15; Journal ID: ISSN 1751-8113

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Feynman integrals; multiple polylogarithms; periods; Chen iterated integrals; elliptic functions

### Citation Formats

```
Henn, Johannes M.
```*Lectures on differential equations for Feynman integrals*. United States: N. p., 2015.
Web. doi:10.1088/1751-8113/48/15/153001.

```
Henn, Johannes M.
```*Lectures on differential equations for Feynman integrals*. United States. doi:10.1088/1751-8113/48/15/153001.

```
Henn, Johannes M. Wed .
"Lectures on differential equations for Feynman integrals". United States.
doi:10.1088/1751-8113/48/15/153001.
```

```
@article{osti_1341773,
```

title = {Lectures on differential equations for Feynman integrals},

author = {Henn, Johannes M.},

abstractNote = {Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space–time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE.},

doi = {10.1088/1751-8113/48/15/153001},

journal = {Journal of Physics. A, Mathematical and Theoretical},

number = 15,

volume = 48,

place = {United States},

year = {Wed Mar 25 00:00:00 EDT 2015},

month = {Wed Mar 25 00:00:00 EDT 2015}

}

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