Determining arbitrary Feynman integrals by vacuum integrals
By introducing an auxiliary parameter, we find a series representation for Feynman integral, which is defined as analytical continuation of a calculable series. To obtain the series representation, one only needs to deal with some much simpler vacuum integrals. The series representation therefore translates the problem of computing Feynman integrals to the problem of performing analytical continuations. As a Feynman integral is fully determined by its series representation, its reduction relation to master integrals can be achieved. Furthermore, differential equations of master integrals with respective to the auxiliary parameter can also be set up. By solving the differential equations, the desired analytical continuations are realized. The series representation thus provides a novel method to reduce and compute Feynman integrals.