Tensor Reduction for Feynman Integrals with Lorentz and Spinor Indices

TL;DR

This paper introduces a graphical method for constructing projectors to reduce complex tensor Feynman integrals with Lorentz and spinor indices in D dimensions, simplifying calculations in quantum field theory.

Contribution

It develops a novel graphical approach and combinatorial formulae for tensor reduction of multi-loop Feynman integrals with Lorentz and spinor indices, including new formulas for spinor basis transformations.

Findings

Constructed projectors for integrals with up to 32 Lorentz indices.

Developed a combinatorial formula linking orbits to integer partitions.

Extended the method to integrals involving spinor indices using gamma matrix properties.

Abstract

We present an efficient graphical approach to construct projectors for the tensor reduction of multi-loop Feynman integrals with both Lorentz and spinor indices in $D$ dimensions. An ansatz for the projectors is constructed making use of its symmetry properties via an orbit partition formula. The graphical approach allows to identify and enumerate the orbits in each case. For the case without spinor indices we find a 1 to 1 correspondence between orbits and integer partitions describing the cycle structure of certain bi-chord graphs. This leads to compact combinatorial formulae for the projector ansatz. With spinor indices the graph-structure becomes more involved, but the method is equally applicable. Our spinor reduction formulae are based on the antisymmetric basis of $\gamma$ matrices, and make use of their orthogonality property. We also provide a new compact formula to pass into the antisymmetric basis. We compute projectors for vacuum tensor Feynman integrals with up to 32 Lorentz indices and up to 4 spinor indices. We discuss how to employ the projectors in problems with external momenta.