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Analytic treatment of the two loop equal mass sunrise graph

S. Laporta, E. Remiddi

TL;DR

The paper delivers an analytic treatment of the equal-mass two-loop sunrise integral across arbitrary momentum transfer by casting the problem into a two-MI differential-equation system and reducing it to a single second-order ODE. It exploits d-dimensional regularization and a Tarasov-style mapping between $d=2$ and $d=4$ to build a bottom-up expansion, solving the homogeneous equation with Euler’s variation of constants using explicitly constructed solutions tied to elliptic integrals. Central results include closed forms for the zeroth-order term $S^{(0)}(2,z)$ and the first-order correction $S^{(1)}(2,z)$ in the $d$-expansion, as well as detailed behavior at all singular points via interpolating and transformed solutions. The methods yield practical representations for fast numerical evaluation and lay groundwork for extending the approach to general-mass two-loop self-energies and precision Standard Model calculations.

Abstract

The two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order differential equation for the scalar Master Integral is expanded in (d-2) and solved by the variation of the constants method of Euler up to first order in (d-2) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.

Analytic treatment of the two loop equal mass sunrise graph

TL;DR

The paper delivers an analytic treatment of the equal-mass two-loop sunrise integral across arbitrary momentum transfer by casting the problem into a two-MI differential-equation system and reducing it to a single second-order ODE. It exploits d-dimensional regularization and a Tarasov-style mapping between and to build a bottom-up expansion, solving the homogeneous equation with Euler’s variation of constants using explicitly constructed solutions tied to elliptic integrals. Central results include closed forms for the zeroth-order term and the first-order correction in the -expansion, as well as detailed behavior at all singular points via interpolating and transformed solutions. The methods yield practical representations for fast numerical evaluation and lay groundwork for extending the approach to general-mass two-loop self-energies and precision Standard Model calculations.

Abstract

The two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order differential equation for the scalar Master Integral is expanded in (d-2) and solved by the variation of the constants method of Euler up to first order in (d-2) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.

Paper Structure

This paper contains 15 sections, 173 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. The differential equations for the MIs.
  3. The expansions in $d$.
  4. The expansion around $d=2$.
  5. Building the solution of the homogeneous equation.
  6. The solutions at the singular points.
  7. The interpolating solutions.
  8. The homogeneous solutions.
  9. Identities under transformations of the argument.
  10. The solution $S^{(0)}(2,z)$ at zeroth order in $(d-2)$.
  11. The solution $S^{(0)}(4,z)$ at zeroth order in $(d-4)$.
  12. The solution $S^{(1)}(2,z)$ at first order in $(d-2)$.
  13. Conclusions.
  14. Relations with the Complete Elliptic Integrals.
  15. Definite Integrals.

Figures (1)

  • Figure 1: The 2-loop sunrise graph.