Low momentum expansion of one loop amplitudes in heterotic string theory

We consider the low momentum expansion of the four graviton and the two graviton-two gluon amplitudes in heterotic string theory at one loop in ten dimensions, and analyze contributions upto the D2ℛ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}^2{\mathrm{\mathcal{R}}}^4 $$\end{document} interaction from the four graviton amplitude, and the D4ℛ2ℱ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}^4{\mathrm{\mathcal{R}}}^2{\mathrm{\mathcal{F}}}^2 $$\end{document} interaction from the two graviton-two gluon amplitude. The calculations are performed by obtaining equations for the relevant modular graph functions that arise in the modular invariant integrals, and involve amalgamating techniques used in the type II theory and the calculation of the elliptic genus in the heterotic theory.