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Commit 0ec39d33 authored by Zhenjie Li's avatar Zhenjie Li
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(**********************************************************************)
(* *)
(* Ancillary file to "The Three-loop MHV Octagon from Qbar equations" *)
(* by Zhenjie Li and Chi Zhang, arXiv:2110.xxxxx. *)
(* *)
(* This file is a replacement rule of all rational letters of the 3 *)
(* loop MHV octagon with the notations: *)
(* ab[i,j,k,l] : Plucker coordinate, *)
(* ab[a,b,c,cap[{d,e},{f,g,h}]] := ab[a,b,c,d] ab[e,f,g,h] *)
(* - ab[a,b,c,e] ab[d,f,g,h], *)
(* ab[a,cap1[{b,c},{d,e},{f,g}]] := ab[a,b,d,e] ab[a,c,f,g] *)
(* - ab[a,b,f,g] ab[a,c,d,e], *)
(* ab[a,b,cap2[{r,s,t},{u,v,w}]] := ab[a,b,s,t] ab[r,u,v,w] *)
(* - ab[a,b,r,t] ab[s,u,v,w] *)
(* + ab[a,b,r,s] ab[t,u,v,w] . *)
(* *)
(**********************************************************************)
{a[1] -> ab[1, 2, 3, 4], a[2] -> ab[1, 2, 3, 5], a[3] -> ab[1, 2, 3, 6],
a[4] -> ab[1, 2, 3, 7], a[5] -> ab[1, 2, 3, 8], a[6] -> ab[1, 2, 4, 5],
a[7] -> ab[1, 2, 4, 6], a[8] -> ab[1, 2, 4, 7], a[9] -> ab[1, 2, 4, 8],
a[10] -> ab[1, 2, 5, 6], a[11] -> ab[1, 2, 5, 7], a[12] -> ab[1, 2, 5, 8],
a[13] -> ab[1, 2, 6, 7], a[14] -> ab[1, 2, 6, 8], a[15] -> ab[1, 2, 7, 8],
a[16] -> ab[1, 3, 4, 5], a[17] -> ab[1, 3, 4, 6], a[18] -> ab[1, 3, 4, 7],
a[19] -> ab[1, 3, 4, 8], a[20] -> ab[1, 3, 5, 6], a[21] -> ab[1, 3, 5, 8],
a[22] -> ab[1, 3, 6, 7], a[23] -> ab[1, 3, 6, 8], a[24] -> ab[1, 3, 7, 8],
a[25] -> ab[1, 4, 5, 6], a[26] -> ab[1, 4, 5, 7], a[27] -> ab[1, 4, 5, 8],
a[28] -> ab[1, 4, 6, 7], a[29] -> ab[1, 4, 6, 8], a[30] -> ab[1, 4, 7, 8],
a[31] -> ab[1, 5, 6, 7], a[32] -> ab[1, 5, 6, 8], a[33] -> ab[1, 5, 7, 8],
a[34] -> ab[1, 6, 7, 8], a[35] -> ab[2, 3, 4, 5], a[36] -> ab[2, 3, 4, 6],
a[37] -> ab[2, 3, 4, 7], a[38] -> ab[2, 3, 4, 8], a[39] -> ab[2, 3, 5, 6],
a[40] -> ab[2, 3, 5, 7], a[41] -> ab[2, 3, 5, 8], a[42] -> ab[2, 3, 6, 7],
a[43] -> ab[2, 3, 6, 8], a[44] -> ab[2, 3, 7, 8], a[45] -> ab[2, 4, 5, 6],
a[46] -> ab[2, 4, 5, 7], a[47] -> ab[2, 4, 5, 8], a[48] -> ab[2, 4, 6, 7],
a[49] -> ab[2, 4, 7, 8], a[50] -> ab[2, 5, 6, 7], a[51] -> ab[2, 5, 6, 8],
a[52] -> ab[2, 5, 7, 8], a[53] -> ab[2, 6, 7, 8], a[54] -> ab[3, 4, 5, 6],
a[55] -> ab[3, 4, 5, 7], a[56] -> ab[3, 4, 5, 8], a[57] -> ab[3, 4, 6, 7],
a[58] -> ab[3, 4, 6, 8], a[59] -> ab[3, 4, 7, 8], a[60] -> ab[3, 5, 6, 7],
a[61] -> ab[3, 5, 6, 8], a[62] -> ab[3, 5, 7, 8], a[63] -> ab[3, 6, 7, 8],
a[64] -> ab[4, 5, 6, 7], a[65] -> ab[4, 5, 6, 8], a[66] -> ab[4, 5, 7, 8],
a[67] -> ab[4, 6, 7, 8], a[68] -> ab[5, 6, 7, 8],
a[69] -> ab[1, 2, cap2[{3, 4, 5}, {6, 7, 8}]],
a[70] -> ab[1, 8, cap2[{2, 3, 4}, {5, 6, 7}]],
a[71] -> ab[2, 3, cap2[{1, 7, 8}, {4, 5, 6}]],
a[72] -> ab[3, 4, cap2[{1, 2, 8}, {5, 6, 7}]],
a[73] -> ab[4, 5, cap2[{1, 2, 3}, {6, 7, 8}]],
a[74] -> ab[5, 6, cap2[{1, 7, 8}, {2, 3, 4}]],
a[75] -> ab[6, 7, cap2[{1, 2, 8}, {3, 4, 5}]],
a[76] -> ab[7, 8, cap2[{1, 2, 3}, {4, 5, 6}]],
a[77] -> ab[1, cap1[{2, 3}, {4, 5}, {6, 7}]],
a[78] -> ab[1, cap1[{2, 3}, {4, 5}, {6, 8}]],
a[79] -> ab[1, cap1[{2, 3}, {4, 5}, {7, 8}]],
a[80] -> ab[1, cap1[{2, 3}, {4, 6}, {7, 8}]],
a[81] -> ab[1, cap1[{2, 3}, {5, 6}, {7, 8}]],
a[82] -> ab[1, cap1[{2, 4}, {5, 6}, {7, 8}]],
a[83] -> ab[1, cap1[{2, 8}, {3, 4}, {5, 6}]],
a[84] -> ab[1, cap1[{2, 8}, {3, 4}, {5, 7}]],
a[85] -> ab[1, cap1[{2, 8}, {3, 4}, {6, 7}]],
a[86] -> ab[1, cap1[{2, 8}, {3, 5}, {6, 7}]],
a[87] -> ab[1, cap1[{2, 8}, {4, 5}, {6, 7}]],
a[88] -> ab[1, cap1[{3, 4}, {5, 6}, {7, 8}]],
a[89] -> ab[2, cap1[{1, 3}, {4, 5}, {6, 7}]],
a[90] -> ab[2, cap1[{1, 3}, {4, 5}, {6, 8}]],
a[91] -> ab[2, cap1[{1, 3}, {4, 5}, {7, 8}]],
a[92] -> ab[2, cap1[{1, 3}, {4, 6}, {7, 8}]],
a[93] -> ab[2, cap1[{1, 3}, {5, 6}, {7, 8}]],
a[94] -> ab[2, cap1[{1, 7}, {3, 4}, {5, 6}]],
a[95] -> ab[2, cap1[{1, 8}, {3, 4}, {5, 6}]],
a[96] -> ab[2, cap1[{1, 8}, {3, 4}, {6, 7}]],
a[97] -> ab[2, cap1[{1, 8}, {3, 5}, {6, 7}]],
a[98] -> ab[2, cap1[{1, 8}, {4, 5}, {6, 7}]],
a[99] -> ab[2, cap1[{3, 4}, {5, 6}, {7, 8}]],
a[100] -> ab[2, cap1[{3, 4}, {5, 7}, {8, 1}]],
a[101] -> ab[3, cap1[{1, 2}, {4, 5}, {6, 7}]],
a[102] -> ab[3, cap1[{1, 2}, {4, 5}, {7, 8}]],
a[103] -> ab[3, cap1[{1, 2}, {4, 6}, {7, 8}]],
a[104] -> ab[3, cap1[{1, 2}, {5, 6}, {7, 8}]],
a[105] -> ab[3, cap1[{1, 7}, {2, 4}, {5, 6}]],
a[106] -> ab[3, cap1[{1, 8}, {2, 4}, {5, 6}]],
a[107] -> ab[3, cap1[{1, 8}, {2, 4}, {5, 7}]],
a[108] -> ab[3, cap1[{1, 8}, {2, 4}, {6, 7}]],
a[109] -> ab[3, cap1[{1, 8}, {4, 5}, {6, 7}]],
a[110] -> ab[3, cap1[{2, 4}, {5, 6}, {7, 8}]],
a[111] -> ab[3, cap1[{2, 8}, {4, 5}, {6, 7}]],
a[112] -> ab[3, cap1[{4, 5}, {6, 8}, {1, 2}]],
a[113] -> ab[4, cap1[{1, 2}, {3, 5}, {6, 7}]],
a[114] -> ab[4, cap1[{1, 2}, {3, 5}, {6, 8}]],
a[115] -> ab[4, cap1[{1, 2}, {3, 5}, {7, 8}]],
a[116] -> ab[4, cap1[{1, 2}, {5, 6}, {7, 8}]],
a[117] -> ab[4, cap1[{1, 3}, {5, 6}, {7, 8}]],
a[118] -> ab[4, cap1[{1, 8}, {2, 3}, {5, 6}]],
a[119] -> ab[4, cap1[{1, 8}, {2, 3}, {5, 7}]],
a[120] -> ab[4, cap1[{1, 8}, {2, 3}, {6, 7}]],
a[121] -> ab[4, cap1[{1, 8}, {3, 5}, {6, 7}]],
a[122] -> ab[4, cap1[{2, 3}, {5, 6}, {7, 8}]],
a[123] -> ab[4, cap1[{2, 8}, {3, 5}, {6, 7}]],
a[124] -> ab[4, cap1[{5, 6}, {7, 1}, {2, 3}]],
a[125] -> ab[5, cap1[{1, 2}, {3, 4}, {6, 7}]],
a[126] -> ab[5, cap1[{1, 2}, {3, 4}, {6, 8}]],
a[127] -> ab[5, cap1[{1, 2}, {3, 4}, {7, 8}]],
a[128] -> ab[5, cap1[{1, 2}, {4, 6}, {7, 8}]],
a[129] -> ab[5, cap1[{1, 3}, {4, 6}, {7, 8}]],
a[130] -> ab[5, cap1[{1, 7}, {2, 3}, {4, 6}]],
a[131] -> ab[5, cap1[{1, 8}, {2, 3}, {4, 6}]],
a[132] -> ab[5, cap1[{1, 8}, {2, 3}, {6, 7}]],
a[133] -> ab[5, cap1[{1, 8}, {2, 4}, {6, 7}]],
a[134] -> ab[5, cap1[{1, 8}, {3, 4}, {6, 7}]],
a[135] -> ab[5, cap1[{2, 3}, {4, 6}, {7, 8}]],
a[136] -> ab[5, cap1[{6, 7}, {8, 2}, {3, 4}]],
a[137] -> ab[6, cap1[{1, 2}, {3, 4}, {5, 7}]],
a[138] -> ab[6, cap1[{1, 2}, {3, 4}, {7, 8}]],
a[139] -> ab[6, cap1[{1, 2}, {3, 5}, {7, 8}]],
a[140] -> ab[6, cap1[{1, 2}, {4, 5}, {7, 8}]],
a[141] -> ab[6, cap1[{1, 7}, {2, 3}, {4, 5}]],
a[142] -> ab[6, cap1[{1, 8}, {2, 3}, {4, 5}]],
a[143] -> ab[6, cap1[{1, 8}, {2, 3}, {5, 7}]],
a[144] -> ab[6, cap1[{1, 8}, {2, 4}, {5, 7}]],
a[145] -> ab[6, cap1[{1, 8}, {3, 4}, {5, 7}]],
a[146] -> ab[6, cap1[{2, 3}, {4, 5}, {7, 8}]],
a[147] -> ab[6, cap1[{2, 8}, {3, 4}, {5, 7}]],
a[148] -> ab[6, cap1[{7, 8}, {1, 3}, {4, 5}]],
a[149] -> ab[7, cap1[{1, 2}, {3, 4}, {5, 6}]],
a[150] -> ab[7, cap1[{1, 2}, {3, 4}, {6, 8}]],
a[151] -> ab[7, cap1[{1, 2}, {3, 5}, {6, 8}]],
a[152] -> ab[7, cap1[{1, 2}, {4, 5}, {6, 8}]],
a[153] -> ab[7, cap1[{1, 3}, {4, 5}, {6, 8}]],
a[154] -> ab[7, cap1[{1, 8}, {2, 3}, {4, 5}]],
a[155] -> ab[7, cap1[{1, 8}, {2, 3}, {4, 6}]],
a[156] -> ab[7, cap1[{1, 8}, {2, 3}, {5, 6}]],
a[157] -> ab[7, cap1[{1, 8}, {3, 4}, {5, 6}]],
a[158] -> ab[7, cap1[{2, 3}, {4, 5}, {6, 8}]],
a[159] -> ab[7, cap1[{2, 8}, {3, 4}, {5, 6}]],
a[160] -> ab[7, cap1[{8, 1}, {2, 4}, {5, 6}]],
a[161] -> ab[8, cap1[{1, 2}, {3, 4}, {5, 6}]],
a[162] -> ab[8, cap1[{1, 2}, {3, 4}, {5, 7}]],
a[163] -> ab[8, cap1[{1, 2}, {3, 4}, {6, 7}]],
a[164] -> ab[8, cap1[{1, 2}, {3, 5}, {6, 7}]],
a[165] -> ab[8, cap1[{1, 2}, {4, 5}, {6, 7}]],
a[166] -> ab[8, cap1[{1, 3}, {4, 5}, {6, 7}]],
a[167] -> ab[8, cap1[{1, 7}, {2, 3}, {4, 5}]],
a[168] -> ab[8, cap1[{1, 7}, {2, 3}, {4, 6}]],
a[169] -> ab[8, cap1[{1, 7}, {2, 3}, {5, 6}]],
a[170] -> ab[8, cap1[{1, 7}, {2, 4}, {5, 6}]],
a[171] -> ab[8, cap1[{1, 7}, {3, 4}, {5, 6}]],
a[172] -> ab[8, cap1[{2, 3}, {4, 5}, {6, 7}]],
a[173] -> ab[cap2[{1, 2, 3}, {3, 4, 5}], cap2[{5, 6, 8}, {7, 8, 1}]],
a[174] -> ab[cap2[{1, 2, 3}, {3, 4, 5}], cap2[{6, 8, 1}, {5, 6, 7}]],
a[175] -> ab[cap2[{1, 2, 7}, {6, 7, 8}], cap2[{2, 3, 5}, {4, 5, 6}]],
a[176] -> ab[cap2[{1, 2, 7}, {6, 7, 8}], cap2[{3, 5, 6}, {2, 3, 4}]],
a[177] -> ab[cap2[{2, 3, 4}, {4, 5, 6}], cap2[{6, 7, 1}, {8, 1, 2}]],
a[178] -> ab[cap2[{2, 3, 4}, {4, 5, 6}], cap2[{7, 1, 2}, {6, 7, 8}]],
a[179] -> ab[cap2[{2, 3, 8}, {7, 8, 1}], cap2[{3, 4, 6}, {5, 6, 7}]],
a[180] -> ab[cap2[{2, 3, 8}, {7, 8, 1}], cap2[{4, 6, 7}, {3, 4, 5}]],
a[181] -> ab[cap2[{3, 4, 1}, {8, 1, 2}], cap2[{4, 5, 7}, {6, 7, 8}]],
a[182] -> ab[cap2[{3, 4, 1}, {8, 1, 2}], cap2[{5, 7, 8}, {4, 5, 6}]],
a[183] -> ab[cap2[{3, 4, 5}, {5, 6, 7}], cap2[{7, 8, 2}, {1, 2, 3}]],
a[184] -> ab[cap2[{3, 4, 5}, {5, 6, 7}], cap2[{8, 2, 3}, {7, 8, 1}]],
a[185] -> ab[cap2[{3, 4, 6}, {5, 6, 7}], cap2[{2, 7, 8}, {1, 2, 3}]],
a[186] -> ab[cap2[{4, 5, 2}, {1, 2, 3}], cap2[{5, 6, 8}, {7, 8, 1}]],
a[187] -> ab[cap2[{4, 5, 6}, {6, 7, 8}], cap2[{1, 3, 4}, {8, 1, 2}]],
a[188] -> ab[cap2[{4, 5, 6}, {6, 7, 8}], cap2[{8, 1, 3}, {2, 3, 4}]],
a[189] -> ab[cap2[{4, 5, 7}, {6, 7, 8}], cap2[{3, 8, 1}, {2, 3, 4}]],
a[190] -> ab[cap2[{5, 6, 3}, {2, 3, 4}], cap2[{6, 7, 1}, {8, 1, 2}]],
a[191] -> ab[cap2[{5, 6, 7}, {7, 8, 1}], cap2[{1, 2, 4}, {3, 4, 5}]],
a[192] -> ab[cap2[{5, 6, 7}, {7, 8, 1}], cap2[{2, 4, 5}, {1, 2, 3}]],
a[193] -> ab[cap2[{5, 6, 8}, {7, 8, 1}], cap2[{4, 1, 2}, {3, 4, 5}]],
a[194] -> ab[cap2[{6, 7, 1}, {8, 1, 2}], cap2[{5, 2, 3}, {4, 5, 6}]],
a[195] -> ab[cap2[{6, 7, 4}, {3, 4, 5}], cap2[{7, 8, 2}, {1, 2, 3}]],
a[196] -> ab[cap2[{6, 7, 8}, {8, 1, 2}], cap2[{2, 3, 5}, {4, 5, 6}]],
a[197] -> ab[cap2[{6, 7, 8}, {8, 1, 2}], cap2[{3, 5, 6}, {2, 3, 4}]],
a[198] -> ab[cap2[{7, 8, 1}, {1, 2, 3}], cap2[{3, 4, 6}, {5, 6, 7}]],
a[199] -> ab[cap2[{7, 8, 1}, {1, 2, 3}], cap2[{4, 6, 7}, {3, 4, 5}]],
a[200] -> ab[cap2[{7, 8, 5}, {4, 5, 6}], cap2[{8, 1, 3}, {2, 3, 4}]],
a[201] -> ab[cap2[{8, 1, 2}, {2, 3, 4}], cap2[{4, 5, 7}, {6, 7, 8}]],
a[202] -> ab[cap2[{8, 1, 2}, {2, 3, 4}], cap2[{5, 7, 8}, {4, 5, 6}]],
a[203] -> ab[cap2[{8, 1, 6}, {5, 6, 7}], cap2[{1, 2, 4}, {3, 4, 5}]],
a[204] -> ab[cap2[{8, 1, 6}, {5, 6, 7}], cap2[{2, 4, 5}, {1, 2, 3}]]}
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