The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter

TL;DR

The paper tackles the analytic computation of the planar double box integral relevant to top-quark pair production with a closed top loop, a two-scale elliptic multi-loop problem not expressible by multiple polylogarithms. By transforming the differential equations for the master integrals into a form linear in $\varepsilon$ with an $\varepsilon^0$ part that is strictly lower triangular, the authors obtain an order-by-order solution in $\varepsilon$ in terms of iterated integrals of elliptic periods and modular forms. They identify three elliptic curves from maximal cuts, show how the solution degenerates to MPLs or to iterated modular forms in particular limits, and construct a 42-element master basis enabling analytic control over the integral. This advances analytic techniques for massive two-loop integrals and provides a framework applicable to other elliptic, multi-scale Feynman diagrams.

Abstract

We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in $\varepsilon$, where the $\varepsilon^0$-part is strictly lower triangular. This system is easily solved order by order in the dimensional regularisation parameter $\varepsilon$. This is an example of an elliptic multi-scale integral involving several elliptic sub-topologies. Our methods are applicable to similar problems.

Figures (4)

• Figure 1: The planar double box. Solid lines correspond to massive propagators of mass $m$, dashed lines correspond to massless propagators. All external momenta are out-going and on-shell: $p_1^2=p_2^2=0$ and $p_3^2=p_4^2=m^2$.
• Figure 2: A sub-topology leading to the square root $\sqrt{-s(-4m^2-s)}$.
• Figure 3: The Feynman graphs associated to the three elliptic curves.
• Figure 4: A sub-topology with an additional relation.