MATAD: a program package for the computation of MAssive TADpoles

TL;DR

MATAD addresses the need for analytic evaluation of massive tadpole integrals at 1- to 3-loop order with a single mass scale. It uses integration-by-parts to reduce integrals to a small set of master integrals and computes their $\varepsilon$-expansions within the FORM-based framework. The package provides a compact input scheme and topology mapping, enabling automation and linkage with diagram generators for complex multi-loop calculations. The examples demonstrate applications to photon polarization, Higgs decay to gluons, and fermion propagators, underscoring MATAD's utility in precision QCD and electroweak corrections.

Abstract

In the recent years there has been an enormous development in the evaluation of higher order quantum corrections. An essential ingredient in the practical calculations is provided by vacuum diagrams, i.e. integrals without external momenta. In this paper a program package is described which can deal with one-, two- and three-loop vacuum integrals with one non-zero mass parameter. The principle structure is introduced and the main parts of the package are described in detail. Explicit examples demonstrate the fields of application.

Figures (11)

• Figure 1: Prototype topologies for one-, two- and three-loop vacuum diagrams. The momentum $p_i$ flows through the line $i$ as indicated by the arrow. Each line may either be massless or carry mass $M$ and may be raised to an arbitrary integer power.
• Figure 2: Flowchart illustrating the structure of MATAD.
• Figure 3: The input of the diagram to be computed can be mapped to one of these topologies. The implemented massive/massless combinations can be found in Appendix \ref{['app:topfile']}. The momentum $p_i$ flows through line $i$ as indicated by the arrow.
• Figure 4: Basic three-loop topologies implemented into MATAD. They are reduced to either simple integrals or master integrals with the help of recurrence relations.
• Figure 5: Master diagram resulting from topology BN1. Here all propagators are raised to power one. Its constant part contains the expression S2 and its ${\cal O}(\varepsilon)$ part OepS2 as listed in Appendix \ref{['app:master']}.