Causality and Loop-Tree Duality at Higher Loops

We relate an $l$-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original $l$-loop graph. Causality requires that the propagators of the trees have a modified $i\delta$ prescription, and we present a simple formula for the correct $i\delta$ prescription.

Figure 1

A two-loop eight-point function with 11 propagators.

Figure 2

A cut diagram corresponding to $\sigma =\{3,9\}$. We also indicate an orientation for the edges $e_3$, $e_5$, $e_6$, and $e_9$.