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

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