General one-loop reduction in generalized Feynman parametrization form

The search for an effective reduction method is one of the main topics in higher loop computation.Recently,an alternative reduction method was proposed by Chen in[1,2].In this paper,we test the power of Chen’s new method using one-loop scalar integrals with propagators of higher power.More explicitly,with the improved version of the method,we can cancel the dimension shift and terms with unwanted power shifting.Thus,the obtained integrating-by-parts relations are significantly simpler and can be solved easily.Using this method,we present explicit examples of a bubble,triangle,box,and pentagon with one doubled propagator.With these results,we complete our previous computations in[3]with the missing tadpole coefficients and show the potential of Chen’s method for efficient reduction in higher loop integrals.