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Leading singularities of Wilson loop correlators from twistor Wilson loop diagrams

James Drummond, Matthew Rochford, Rowan Wright

Abstract

The leading singularities of one-loop scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory are known to factorise into products of tree-level amplitudes, and this can be seen from a number of different perspectives e.g. generalised unitarity or on-shell diagrams. Here we investigate the leading singularities from the perspective of the Wilson loop expectation values to which these amplitudes are dual, in particular making use of the twistor Wilson loop formalism. We show that the factorisation of one-loop leading singularities of a null Wilson loop's expectation value into a product of tree-level objects is manifest at the level of twistor Wilson loop diagrams, and is a simple consequence of planarity, without appeal to e.g. unitarity on the amplitude side of the duality. We then use the same approach to derive compact formulae for the one-loop leading singularities of correlators of multiple light-like Wilson loop operators in terms of tree-level objects. Via the chiral box expansion, these formulae provide a simple route to writing down the $O(g^2)$ correlation function of any number of Wilson loops at any MHV degree.

Leading singularities of Wilson loop correlators from twistor Wilson loop diagrams

Abstract

The leading singularities of one-loop scattering amplitudes in planar super Yang-Mills theory are known to factorise into products of tree-level amplitudes, and this can be seen from a number of different perspectives e.g. generalised unitarity or on-shell diagrams. Here we investigate the leading singularities from the perspective of the Wilson loop expectation values to which these amplitudes are dual, in particular making use of the twistor Wilson loop formalism. We show that the factorisation of one-loop leading singularities of a null Wilson loop's expectation value into a product of tree-level objects is manifest at the level of twistor Wilson loop diagrams, and is a simple consequence of planarity, without appeal to e.g. unitarity on the amplitude side of the duality. We then use the same approach to derive compact formulae for the one-loop leading singularities of correlators of multiple light-like Wilson loop operators in terms of tree-level objects. Via the chiral box expansion, these formulae provide a simple route to writing down the correlation function of any number of Wilson loops at any MHV degree.
Paper Structure (20 sections, 94 equations, 17 figures)

This paper contains 20 sections, 94 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Review of background material
  3. The twistor Wilson loop
  4. Tree level calculations
  5. Calculations at O($g^2$)
  6. Schubert problems and Schubert solutions
  7. Leading singularities for a single Wilson loop
  8. Four mass leading singularities
  9. Lower mass example: two mass easy
  10. Leading singularities for multiple Wilson Loop correlators
  11. 2-2 split
  12. Adding spectator Wilson loops
  13. 4- split
  14. 3-1 split
  15. 2-1-1 split
  16. ...and 5 more sections

Figures (17)

  • Figure 1: An example of a planar diagram which contributes to the correlator of two Wilson loops at N$^3$MHV.
  • Figure 2: An example of a planar diagram which contributes at N$^2$MHV to the connected part of the correlation function of two Wilson loops. Here $i_1$ and $i_2$ label cusps on one Wilson loop, and $j$ labels a cusp on another Wilson loop
  • Figure 3: A planar twistor diagram which contributes at N${}^2$MHV to a pentagon-pentagon correlator at O($g^2$).
  • Figure 4: A box diagram drawn for the three-mass cut $(12)(23)(45)(67)$.
  • Figure 5: The only planar twistor diagram for a single Wilson loop expectation value which has non-zero residue on the quadruple pole in Eq. \ref{['quadruplePole']}.
  • ...and 12 more figures