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Massive On-shell Recursion Relations for n-point Amplitudes

We construct two and three-line shifts for tree-level amplitude with massless and/or massive particles, and provide a method to construct general multi-line shifts for all masses. We choose the massless-massive BCFW shift from these shifts and examine its validity in renormalizable theories. Using such a shift, we find that amplitudes with at least one massless vector boson are constructible. This reveals the importance of gauge theory in the construction of amplitudes with massive particles. We also find that this kind of amplitudes have a cancellation related to group structure among different channels, which is essential for constructibility. Furthermore, we show that in the limit of large shift parameter $z$, the amplitude with four massive vector bosons, which can include transverse massive vector particles, have structures proportional to the amplitude with shifted vector particles replaced by Goldstone bosons in the leading order. This is responsible for the failure of massive-massive BCFW recursion relations in the amplitudes with four massive vector bosons.