Analytic results for the planar double box integral relevant to top-pair production with a closed top loop

TL;DR

The paper delivers a comprehensive analytic treatment of the planar double box integral for top-pair production with a closed top loop by recasting its master-integral system into an $\varepsilon$-form differential-equation framework. It reveals that three inequivalent elliptic curves govern the topology, extracted from maximal cuts, and expresses the master integrals as iterated integrals over a rich set of integration kernels including polylogarithmic and modular-form components. The authors provide explicit expansions up to $\varepsilon^4$ for 44 master integrals, along with boundary constants and extensive cross-checks against numerical sector-decomposition results. This work extends analytic multi-loop techniques to multi-scale problems with multiple elliptic curves and demonstrates a path toward broader applicability in NNLO computations for collider phenomenology.

Abstract

In this article we give the details on the analytic calculation of the master integrals for the planar double box integral relevant to top-pair production with a closed top loop. We show that these integrals can be computed systematically to all order in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the system of differential equations into a form linear in $\varepsilon$, where the $\varepsilon^0$-part is a strictly lower triangular matrix. Explicit results in terms of iterated integrals are presented for the terms relevant to NNLO calculations.

Figures (6)

• 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: The auxiliary topology with nine propagators.
• Figure 3: Master topologies (part 1).
• Figure 4: Master topologies (part 2).
• Figure 5: Master topologies (part 3).