Seven-Point Amplitudes in Planar N=4 SYM
Computer-readable ancillary files associated with the paper
"Lifting Heptagon Symbols to Functions''
arXiv:2007.12966
by Lance Dixon and Yu-Ting (Andy) Liu

The Origin and the CO surface
  • The MHV remainder function and the NMHV ratio function components, evaluated at the origin and on the CO surface, from 1 loop through 4 loops in terms of one-dimensional harmonic polylogarithms (HPLs) (2.4MB): R_P_o_co.txt

    • Amplitudes Defined via Coproducts
  • All the definitions needed to specify the MHV and NMHV amplitudes through 4 loops. We do so via the amplitudes' differentials, or {n-1,1} coproducts, plus their values on the CO surface. As part of this description, we provide a complete basis for heptagon functions through weight 6 (51MB unzipped): HCoproductTables.txt.zip
  • Weight "wt" parity even functions are denoted by HE[wt,i], and parity odd functions by HO[wt,i].

    • And More Coproducts
  • To go to 4 loops, we also need to provide a "trimmed" coproduct table at weight 7, mapping to the full weight 6 function space, for the 154 parity-even {7,1} coproducts of the MHV and NMHV 4 loop amplitudes (134MB unzipped): MHE_7.txt.zip
  • and similarly for the 154 parity-odd {7,1} coproducts (82MB unzipped): MHO_7.txt.zip
  • and similarly for the 16 weight 8 parity-even parts of the 4 loop amplitudes themselves, mapping to weight 7 (62KB): MHE_8.txt
  • and similarly for the 15 weight 8 parity-odd parts (61KB): MHO_8.txt

  • The total number of functions described by these tables at weight 1,2,...,8 is:
  • HE: 7, 28, 92, 288, 828, 2284, 154, 16
  • HO: 0, 0, 6, 28, 120, 412, 154, 15

  • Expressions for the MHV and NMHV amplitudes through 4 loops in terms of the HE and HO functions (0.8MB): AmpsH.txt
    • Dihedral symmetries
  • Action of the cyclic and flip generators of the dihedral group D7 on the heptagon functions (36MB unzipped): HDihedralSym.txt.zip

    • Values of all the heptagon functions on the CO surface
  • As boundary values for integrating up any function, we provide the values on the CO surface of all heptagon functions through weight 6, and the "trimmed" ones at weights 7 and 8 (43MB unzipped): HcoTable.txt.zip

    • Only 406 weight 6 parity odd combinations?
  • The odd parts of the 3 loop amplitudes and 4 loop amplitudes' {6,1,1} coproducts span only a 406 dimensional space, where the coefficients of all the beyond-the-symbol functions are fixed, once the symbol-level function is specified. We provide this subspace in terms of the HO[6,i] functions (35KB): weight6odd406.txt

    • Links to Six-Point Results
  • Six-point amplitudes through 7 loops, and cosmically normalized Steinmann satisfying hexagon functions