Direct Expression for One-Loop Tensor Reduction with Lorentz Indices via Generating Function

Abstract

In recent work, we derived a direct expression for one-loop tensor reduction using generating functions and Feynman parametrization in projective space, avoiding recursive relations. However, for practical applications, this expression still presents two challenges: (1) While the final reduction coefficients are expressed in terms of the dimension D and Mandelstam variables, the given expression explicitly contains irrational functions; (2) The expression involves an auxiliary vector R, which can be eliminated via differentiation $\frac{\partial}{\partial R}$, but the presence of irrational terms making differentiation cumbersome. (3) Most practical applications require the tensor form with Lorentz indices. In this paper, we provide a rational form of the reduction coefficients with Lorentz indices, free from recursion. Additionally, We provide a pure Wolfram Mathematica implementation of the code. Our practical tests demonstrate that this direct expression achieves significantly higher computational efficiency compared to the traditional Passarino-Veltman (PV) reduction or other recursion-based methods.