Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter

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 regularization parameter $ϵ$. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in $ϵ$, where the ${ϵ}^0$ part is strictly lower triangular. This system is easily solved order by order in the dimensional regularization parameter $ϵ$. This is an example of an elliptic multiscale integral involving several elliptic subtopologies. Our methods are applicable to similar problems.

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 subtopology leading to the square root $\sqrt{-s(-4m^2-s)}$.

Figure 3

The Feynman graphs associated to the three elliptic curves.

Figure 4

A subtopology with an additional relation.