Extended projection method for massive fermions

Tensor reduction is of considerable importance in calculations of multi-loop amplitudes, and the projection method is one of the most popular approaches for tensor reduction. However, the projection method can be problematic when applied to amplitudes with massive fermions, due to the inconsistency between helicity and chirality.We propose an extended projection method for reducing the loop amplitude which contains a fermion chain with two massive spinors. The extension is achieved by decomposing one of the massive spinors into two massless spinors, the"null spinor" and the "reference spinor". The extended projection method can be effectively applied in all processes,including the production of massive fermions. We present the tensor reduction for a virtual Z boson decaying into a top-quark pair as a demonstration of our approach.