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Up until now, the BCFW technique has been a widely used method in getting theamplitudes in various theories. Usually, the vanishing of the boundary term isnecessary for the efficiency of the method. However, there are also many kindsof amplitudes which will have boundary terms. Hence it will be nice if it ispossible to get the boundary terms in an efficient manner. As is well-known, ingauge theory the Ward identity imposes constraints on the form of theamplitude. In [1], we studied the Ward identity with a pair of shifted linesand the implied recursion relations. In this article, we discuss thecomplexified Ward identity in more detail. In particular we give a proof of thecomplexified Ward identity directly from the Feynman rules in Feynman-Lorentzgauge. Furthermore, we give more examples in calculating the one-line off-shellcurrents using this technique.

Complexified Ward Identity in pure Yang-Mills theory at tree-level