Direct expression for one-loop tensor reduction with Lorentz indices via a generating function

In [], 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mfrac><a:mo stretchy="false">∂</a:mo><a:mrow><a:mo stretchy="false">∂</a:mo><a:mi>R</a:mi></a:mrow></a:mfrac></a:math>, but the presence of irrational terms making differentiation cumbersome; and 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 implementation of the code. Our practical tests demonstrate that this direct expression achieves significantly higher computational efficiency compared to the traditional Passarino-Veltman reduction or other recursion-based methods.