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A new method for taming tensor sum-integrals

Ioan Ghisoiu, York Schröder

Abstract

We report on the computation of a class of massless bosonic three-loop vacuum sum-integrals which are key building blocks for an evaluation of the Debye screening mass in hot QCD. Generalizing known techniques and introducing the concept of tensor reduction by dimensionality shifts (known to the zero-temperature community since the work of Tarasov in 1996) to finite temperature, we are able to treat hitherto unaccessible cases, which will allow us to finalize the long-term project of NNLO Debye mass evaluation.

A new method for taming tensor sum-integrals

Abstract

We report on the computation of a class of massless bosonic three-loop vacuum sum-integrals which are key building blocks for an evaluation of the Debye screening mass in hot QCD. Generalizing known techniques and introducing the concept of tensor reduction by dimensionality shifts (known to the zero-temperature community since the work of Tarasov in 1996) to finite temperature, we are able to treat hitherto unaccessible cases, which will allow us to finalize the long-term project of NNLO Debye mass evaluation.

Paper Structure

This paper contains 27 sections, 73 equations.

Table of Contents

  1. Introduction
  2. Decomposition of ${\cal M}_{N,-2}$
  3. Splitting off trivial parts
  4. Taming the tensors
  5. Decomposition of spectacles-type sum-integrals
  6. Non-zero modes
  7. Generic formulae for 2-point subtraction terms
  8. Generic formulae for divergences $V^{\rm d}$ of non-zero modes
  9. Specific results for divergences $V^{\rm d}$ of non-zero modes
  10. Finite parts $V^{\rm f}$ of non-zero modes
  11. Zero modes
  12. Generic results for divergences $S^{\rm d}$ of zero modes
  13. Specific results for divergences $V^{\rm z,d}$ of zero modes
  14. Finite parts $V^{\rm z,f}$ of zero modes
  15. Result for ${\cal M}_{3,-2}$
  16. ...and 12 more sections