Contraction and decomposition matrices for vacuum diagrams

TL;DR

This paper introduces recurrence relations and explicit formulas for contraction and decomposition matrices used in tensor reduction of vacuum diagrams, including a detailed 3-loop case and its determinant.

Contribution

It provides the first general recurrence relations and explicit formulas for these matrices, advancing tensor reduction techniques for vacuum diagrams.

Findings

Derived general recurrence relations for contraction and decomposition matrices.

Presented an explicit formula for the 3-loop decomposition matrix.

Calculated the determinant of the 3-loop decomposition matrix.

Abstract

Tensor reduction of vacuum diagrams uses contraction and decomposition matrices. We present general recurrence relations for the calculation of those matrices and an explicit formula for the 3-loop decomposition matrix and its determinant.