Boundary Contributions of On-shell Recursion Relations With Multiple-line Deformation

On-shell recursion relation has been recognized as a powerful tool for calculating tree level amplitudes in quantum field theory, but it doesn't work well when the residue of the deformed amplitude $\hat{A}(z)$ doesn't vanish at infinity of $z$. However, in such situation, we still can get the right amplitude by computing the boundary contribution explicitly. In arXiv:0801.2385, background field method was first used to analyze the boundary behaviors of amplitudes with two deformed external lines in different theories. The same method has also been generalized to calculate the explicit boundary operators of some amplitudes with BCFW-like deformation in arXiv:1507.00463. In this paper, we will take a step further to generalize the method into the case of multiple-line deformation, and to show how the boundary behaviors (even the boundary contributions) can be extracted in the method.