The unequal mass sunrise integral expressed through iterated integrals on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mo>‾</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:math>
Christian Bogner, Stefan Müller–Stach, Stefan Weinzierl
Abstract
We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter epsilon. In order to do so, we transform the system of differential equations for the master integrals to an epsilon-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M-1,M-3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on (M) over bar (1,3). On the hypersurface tau = constour result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms. (C) 2020 The Author(s). Published by Elsevier B.V.