# The Mathematics of Linear Relations between Feynman Integrals

18-22 March 2019
Mainz Institute for Theoretical Physics, Johannes Gutenberg University
Europe/Berlin timezone

## L-loop watermelon and sunrise graphs: differential equations, index and dimension shifting relations, and quadratic constraints

21 Mar 2019, 09:30
1h
02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

### 02.430

#### Mainz Institute for Theoretical Physics, Johannes Gutenberg University

Staudingerweg 9 / 2nd floor, 55128 Mainz

### Speaker

Dr Roman Lee (Budker Institute of Nuclear Physics, Novosibirsk)

### Description

We derive reduction formulae, differential equations and dimensional recurrence relations for $L$-loop two-point massive tadpoles and sunrises with arbitrary masses and regularized both dimensionally and analytically. The differential system obtained has a Pfaff form and can be turned into $(\epsilon+\tfrac{1}{2})$-form when the analytic regularization is removed. For odd $d$ this form allows us to present coefficients of $\epsilon$-expansion explicitly in terms of Goncharov's polylogarithms. Using the symmetry properties of the matrix in the right-hand side of the differential system, we obtain quadratic constraints for the solutions of the obtained differential system near any integer $d$.

### Primary author

Dr Roman Lee (Budker Institute of Nuclear Physics, Novosibirsk)