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Analytic Continuation of Massless Two-Loop Four-Point Functions

T. Gehrmann, E. Remiddi

TL;DR

This work provides a comprehensive framework to analytically continue massless two-loop four-point functions with one off-shell leg from the Euclidean region to all physically relevant Minkowski regions. By expressing master integrals in terms of harmonic polylogarithms (HPLs) and two-dimensional harmonic polylogarithms (2dHPLs), the authors derive explicit continuation formulas and an algorithmic procedure to map arbitrary kinematics into analytically tractable forms. They detail region-by-region transformations, including real and imaginary parts, and present weight-by-weight strategies to extend beyond weight-1, enabling numerical evaluation across all channels. The results open the way to derive two-loop master integrals and unrenormalized matrix elements for hadronic vector-boson-plus-jet production and DIS$(2+1)$-jet production from previous three-jet results, with practical implications for NNLO jet phenomenology.

Abstract

We describe the analytic continuation of two-loop four-point functions with one off-shell external leg and internal massless propagators from the Euclidean region of space-like $1\to 3$ decay to Minkowskian regions relevant to all $1\to 3$ and $2\to 2$ reactions with one space-like or time-like off-shell external leg. Our results can be used to derive two-loop master integrals and unrenormalized matrix elements for hadronic vector-boson-plus-jet production and deep inelastic two-plus-one-jet production, from results previously obtained for three-jet production in electron--positron annihilation.

Analytic Continuation of Massless Two-Loop Four-Point Functions

TL;DR

This work provides a comprehensive framework to analytically continue massless two-loop four-point functions with one off-shell leg from the Euclidean region to all physically relevant Minkowski regions. By expressing master integrals in terms of harmonic polylogarithms (HPLs) and two-dimensional harmonic polylogarithms (2dHPLs), the authors derive explicit continuation formulas and an algorithmic procedure to map arbitrary kinematics into analytically tractable forms. They detail region-by-region transformations, including real and imaginary parts, and present weight-by-weight strategies to extend beyond weight-1, enabling numerical evaluation across all channels. The results open the way to derive two-loop master integrals and unrenormalized matrix elements for hadronic vector-boson-plus-jet production and DIS-jet production from previous three-jet results, with practical implications for NNLO jet phenomenology.

Abstract

We describe the analytic continuation of two-loop four-point functions with one off-shell external leg and internal massless propagators from the Euclidean region of space-like decay to Minkowskian regions relevant to all and reactions with one space-like or time-like off-shell external leg. Our results can be used to derive two-loop master integrals and unrenormalized matrix elements for hadronic vector-boson-plus-jet production and deep inelastic two-plus-one-jet production, from results previously obtained for three-jet production in electron--positron annihilation.

Paper Structure

This paper contains 13 sections, 132 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Kinematic regions and notation
  3. Basics of analytic continuation
  4. Analytic continuation in one invariant for time-like $q^2$
  5. Analytic continuation in one invariant for space-like $q^2$
  6. Continuation from (1a)$_-$ to (1d)$_-$ and to (1b)$_-$, (1c)$_-\ $
  7. Continuation from (1a)$_-$ to (4d)$_-$ and to (2d)$_-$,(3d)$_-$
  8. Continuation from (1a)$_-$ to (4b)$_-$ and to (2b)$_-$, (3b)$_-$
  9. Continuation from (1a)$_-$ to (3c)$_-$ and to (4c)$_-$ and (2c)$_-$
  10. Conclusions
  11. Harmonic polylogarithms
  12. One-dimensional harmonic polylogarithms
  13. Two-dimensional harmonic polylogarithms

Figures (3)

  • Figure 1: Kinematic plane in terms of Lorentz invariants $s_{ij} = (p_i+p_j)^2$ displayed in equilateral coordinates.
  • Figure 2: Regions of the kinematic plane relevant to (a) $e^+e^- \to 3$ jets $(q^2>0)$, (b) $(V+ 1)$ jet production $(q^2>0)$ and (c) deep inelastic $(2+1)$ jet production $(q^2<0)$.
  • Figure 3: Kinematic plane in Cartesian coordinates.