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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 (11 sections, 25 equations, 4 figures)

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: