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The one-loop six-dimensional hexagon integral with three massive corners

Vittorio Del Duca, Lance J. Dixon, James M. Drummond, Claude Duhr, Johannes M. Henn, Vladimir A. Smirnov

TL;DR

Addresses the one-loop six-dimensional hexagon integral with three non-adjacent masses, deriving differential equations to relate it to known pentagons, computing the symbol, and reconstructing the full weight-3 function. By using a momentum-twistor parametrization that makes the discriminant a perfect square, the authors obtain a compact analytic form as a 24-term sum built from a single weight-three master function. The result is expressed in terms of logarithms and polylogarithms of six dual conformal cross-ratios and is numerically validated against a parametric integral in the relevant region. The work connects six-dimensional hexagon integrals to dual-conformal structures in N=4 SYM and provides a framework for understanding higher-loop amplitude/Wilson-loop structures.

Abstract

We compute the six-dimensional hexagon integral with three non-adjacent external masses analytically. After a simple rescaling, it is given by a function of six dual conformally invariant cross-ratios. The result can be expressed as a sum of 24 terms involving only one basic function, which is a simple linear combination of logarithms, dilogarithms, and trilogarithms of uniform degree three transcendentality. Our method uses differential equations to determine the symbol of the function, and an algorithm to reconstruct the latter from its symbol. It is known that six-dimensional hexagon integrals are closely related to scattering amplitudes in N=4 super Yang-Mills theory, and we therefore expect our result to be helpful for understanding the structure of scattering amplitudes in this theory, in particular at two loops.

The one-loop six-dimensional hexagon integral with three massive corners

TL;DR

Addresses the one-loop six-dimensional hexagon integral with three non-adjacent masses, deriving differential equations to relate it to known pentagons, computing the symbol, and reconstructing the full weight-3 function. By using a momentum-twistor parametrization that makes the discriminant a perfect square, the authors obtain a compact analytic form as a 24-term sum built from a single weight-three master function. The result is expressed in terms of logarithms and polylogarithms of six dual conformal cross-ratios and is numerically validated against a parametric integral in the relevant region. The work connects six-dimensional hexagon integrals to dual-conformal structures in N=4 SYM and provides a framework for understanding higher-loop amplitude/Wilson-loop structures.

Abstract

We compute the six-dimensional hexagon integral with three non-adjacent external masses analytically. After a simple rescaling, it is given by a function of six dual conformally invariant cross-ratios. The result can be expressed as a sum of 24 terms involving only one basic function, which is a simple linear combination of logarithms, dilogarithms, and trilogarithms of uniform degree three transcendentality. Our method uses differential equations to determine the symbol of the function, and an algorithm to reconstruct the latter from its symbol. It is known that six-dimensional hexagon integrals are closely related to scattering amplitudes in N=4 super Yang-Mills theory, and we therefore expect our result to be helpful for understanding the structure of scattering amplitudes in this theory, in particular at two loops.

Paper Structure

This paper contains 8 sections, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Integral representation and differential equations
  3. Symbols from differential equations
  4. Twistor geometry associated to a three-mass hexagon
  5. Integrating the symbol: the one-loop three-mass hexagon integral
  6. Conclusion
  7. Special cases
  8. Arguments in terms of space-time cross ratios

Figures (2)

  • Figure 1: (a) depicts the representation of $H_{9}$ as a line integral, see Eqs. (\ref{['H9line']}) and (\ref{['H9line2']}). The differential operator in Eq. (\ref{['diffeqH9']}) localizes the $y_{1}$ integration to $x_{2}$, yielding $P_{8}(x_{2},x_{4},x_{5},x_{7},x_{8})$, see (b).
  • Figure 2: The one-loop three-mass hexagon integral (left), and its geometric configuration in momentum twistor space $\mathbb{CP}^3$ (right). Only the intersection points $Z_1$, $Z_4$ and $Z_7$ have an intrinsic geometrical meaning, whereas all other twistors can be moved freely along the lines.