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ON-SHELL2: FORM based package for the calculation of two-loop self-energy single scale Feynman diagrams occurring in the Standard Model

J. Fleischer, M. Yu. Kalmykov

TL;DR

The paper presents ON-SHELL2, a FORM-based package for calculating two-loop self-energy diagrams with a single non-zero internal mass and on-shell external momentum. It develops a comprehensive integration-by-parts recurrence framework, organized by F-, V-, and J-topologies, to reduce arbitrary-index diagrams to a set of master integrals, with master-integral analytics compiled via differential equations. The package provides explicit FORM procedures, a DIANA-generated workflow, and demonstrates on-shell results including ε-pole structures and mass-difference expansions. This work extends existing tools like SHELL2 and enables efficient, high-precision radiative correction calculations in the Standard Model. The approach improves robustness against ε-expansions and broadens the range of calculable on-shell two-loop self-energies in the SM.

Abstract

A FORM based package (ON-SHELL2) for the calculation of two loop self-energy diagrams with one nonzero mass in internal lines and the external momentum on the same mass shell is elaborated. The algorithm, based on recurrence relations obtained from the integration-by-parts method, allows us to reduce diagrams with arbitrary indices (powers of scalar propagators) to a set of master integrals. The SHELL2 package is used for the calculation of special types of diagrams. Analytical results for master integrals are collected.

ON-SHELL2: FORM based package for the calculation of two-loop self-energy single scale Feynman diagrams occurring in the Standard Model

TL;DR

The paper presents ON-SHELL2, a FORM-based package for calculating two-loop self-energy diagrams with a single non-zero internal mass and on-shell external momentum. It develops a comprehensive integration-by-parts recurrence framework, organized by F-, V-, and J-topologies, to reduce arbitrary-index diagrams to a set of master integrals, with master-integral analytics compiled via differential equations. The package provides explicit FORM procedures, a DIANA-generated workflow, and demonstrates on-shell results including ε-pole structures and mass-difference expansions. This work extends existing tools like SHELL2 and enables efficient, high-precision radiative correction calculations in the Standard Model. The approach improves robustness against ε-expansions and broadens the range of calculable on-shell two-loop self-energies in the SM.

Abstract

A FORM based package (ON-SHELL2) for the calculation of two loop self-energy diagrams with one nonzero mass in internal lines and the external momentum on the same mass shell is elaborated. The algorithm, based on recurrence relations obtained from the integration-by-parts method, allows us to reduce diagrams with arbitrary indices (powers of scalar propagators) to a set of master integrals. The SHELL2 package is used for the calculation of special types of diagrams. Analytical results for master integrals are collected.

Paper Structure

This paper contains 11 sections, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The recurrence relations.
  3. F-topology
  4. V-topology
  5. J-topology
  6. Master-integrals
  7. Use of the package
  8. Conclusion
  9. The set of recurrence relations for F-type integrals
  10. The set of recurrence relations for V-type integrals
  11. Analytical results

Figures (4)

  • Figure 1: The F, V and J topologies. Bold and thin lines correspond to the mass and massless propagators, respectively.
  • Figure 2: Notations used in present paper.
  • Figure 3: "Triangle" rule.
  • Figure 4: