On multi-step BCFW recursion relations

In this paper, we extensively investigate the new algorithm known as the multi- step BCFW recursion relations. Many interesting mathematical properties are found and understanding these aspects, one cannd a systematic way to complete the calculation of amplitude afternite, denite steps and get the correct answer, without recourse to any specic knowledge fromeld theories, besides mass dimension and helicities. This process consists of the pole concentration and inconsistency elimination. Terms that sur- vive inconsistency elimination cannot be determined by the new algorithm. They include polynomials and their generalizations, which turn out to be useful objects to be explored. Afterwards, we apply it to the Standard Model plus gravity to illustrate its power and lim- itation. Ensuring its workability, we also tentatively discuss how to improve its eciency by reducing the steps.