The four-loop ratio function
Computer-readable files associated with the paper
"The four-loop six-gluon NMHV ratio function''
arXiv:1509.08127
by Lance Dixon, Matt von Hippel and Andrew McLeod

Functional Integrability Relations
  • Linear relations among the {n-2,1,1} coproduct components of a weight n hexagon function F: Integrability Relations

    • Ratio and Remainder Functions
  • We use a basis of hexagon functions through weight 8 to describe the ratio function and the remainder function through four loops. We actually give the NMHV amplitude divided by the BDS-like ansatz, both the parity-even part E6 (with coefficients E61, E62, E63, E64) and the parity odd part Et6 (with coefficients Et62, Et63, Et64). For the (MHV) remainder function R6 we give R62, R63, R64. We also give the conversion relations between (E6,Et6) and the ratio function componenents (V,Vt): EEtR6

    • Hexagon Function Basis
  • We give a basis of irreducible hexagon functions through weight 8. We describe the basis iteratively, by listing the {n-1,1} coproduct components for each weight n function in terms of the weight n-1 functions.
  • Parity-odd weight 6 Parity-even weight 6
  • Parity-odd weight 7 Parity-even weight 7
  • Parity-odd weight 8 Parity-even weight 8

    • Near collinear limit
  • The expansion of V4 and Vt4 (and the lower-loop coefficients) in the near collinear limit w -> 0, as a function of T, F and S, through order T^3 (gzipped): NMHV4LT3

    • Multi-Regge limit
  • The transcendental functions p_n^{(L)} and q_n^{(L)} are given through four loops in terms of single-valued harmonic polylogarithms (SVHPLs) NMHV4LMRK

    • Multiple Polylog Representation
  • We give the ratio and remainder functions through four loops in terms of multiple polylogarithms (G functions) in two bulk regions inside the unit cube:
  • Region I
  • Region II

    • The lines (u,u,1), (u,1,u), (u,1,1) and (1,u,1)
  • The ratio function and remainder function on these lines, in terms of one-dimensional harmonic polylogarithms (HPLs): NMHV4LLines

    • Spurious Pole Surface Functions
  • We give a basis for the functions f(u,v) on the spurious pole surface w=1 through weight 7. We describe the basis iteratively, by listing the {n-1,1} coproduct components for each weight n function in terms of the weight n-1 functions.
  • SpuriousPoleSurfaceBasis

  • We also give the ratio function and remainder function on the spurious pole surface w=1 through three loops, using this function basis.
  • SpuriousPoleSurfaceEEtR6

  • Similar files for the four-loop remainder function

  • An older page for the three-loop ratio function