Top-forms of Leading Singularities in Nonplanar Multi-loop Amplitudes

The on-shell diagram is a very important tool in studying scattering amplitudes. In this paper we discuss the on-shell diagrams without external BCFW-bridges. We introduce an extra step of adding an auxiliary external momentum line. Then we can decompose the on-shell diagrams by removing external BCFW-bridges to a planar diagram whose top-form is well-known to all now. The top-form of the on-shell diagram with the auxiliary line can be obtained by adding the BCFW-bridges in an inverse order as discussed in our former paper \cite{Chen:2014ara}. To get the top-form of the original diagram, the soft limit of the auxiliary line is needed. We obtain the evolution rule for the Grassmannian integral and the geometry constraint under the soft limit. This completes the top-form description of leading singularities in nonplanar scattering amplitudes of $\mathcal{N}=4$ Super Yang-Mills (SYM), which is valid for arbitrary higher-loops and beyond the Maximally-Helicity-Violation (MHV) amplitudes.