Boundary contributions of on-shell recursion relations with multiple-line deformation

The on-shell recursion relation has been recognized as a powerful tool for calculating tree-level amplitudes in quantum field theory, but it does not work well when the residue of the deformed amplitude $$\hat{A}(z)$$ does not vanish at infinity of z. However, in such a situation, we still can get the right amplitude by computing the boundary contribution explicitly. In Arkani-Hamed and Kaplan (JHEP 04:076. https://doi.org/10.1088/1126-6708/2008/04/076 . arXiv:0801.2385 , 2008), the 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 been generalized to calculate the explicit boundary operators of some amplitudes with BCFW-like deformation in Jin and Feng (JHEP 04:123. https://doi.org/10.1007/JHEP04(2016)123 . arXiv:1507.00463 , 2016). In this paper, we will take a step further to generalize the method to the case of multiple-line deformation, and to show how the boundary behaviors (even the boundary contributions) can be extracted in the method.