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Boundary Behaviors for General Off-shell Amplitudes in Yang-Mills Theory

In this article, we analyze the boundary behaviors of pure Yang-Mills amplitudes under adjacent and non adjacent BCFW shifts in Feynman gauge. We introduce reduced vertexes for Yang-Mills fields, prove that these reduced vertexes are equivalent to the original vertexes as for the study of boundary behaviors, which greatly simplifies our analysis of boundary behaviors. Boundary behaviors for adjacent shifts are readily obtained using reduced vertexes. Then we prove a theorem on permutation sum and use it to prove the improved boundary behaviors for non-adjacent shifts. Based on the boundary behaviors, we find that it is possible to generalize BCFW recursion relation to calculate general tree level off shell amplitudes.