BGap.h

   1 * This file implements the Berends-Giele recursion to compute the
   2 * alpha' expansion of disk integrals using the method described in the
   3 * paper arXiv:1609.07078
   4 *
   5 * Non-abelian Z-theory: Berends-Giele recursion for the alpha' expansion
   6 * of disk integrals
   7 *
   8 * by Carlos R. Mafra and Oliver Schlotterer.
   9 *
  10 * You can use this file as you wish, as long as you cite the
  11 * paper arXiv:1609.07078 afterwards. In addition, consider citing
  12 * the paper arXiv:1603.09731 as well, as it is very closely related
  13 *
  14 * Copyright (c) 2016 Carlos R. Mafra
  15
  16         Indices a,b,c,d,e,f,g,h,i,j,k,l;
  17         Indices m,n,p,q,r,ss,t,u,v,x;
  18         Indices <m1,n1>,...,<m20,n20>;
  19         Indices i1,...,i20;
  20         Indices j1,...,j20;
  21
  22         Symbols [p],[c];
  23         Symbol x1;
  24         Symbols <zeta2>,...,<zeta10>;
  25
  26 * Mandelstam s(i,j), Is(i,j)=1/s(i,j), KK(i,[c],j) = ki.kj
  27         CFunction s(s), Is(s), KK;
  28
  29         CFunction phi, Zint;
  30         CFunction nargs;
  31
  32         CFunction ctmpF1,countL,countR;
  33         Function tmpF, tmpF1;
  34
  35 * phi(i1,i2,i3|j1,j2,j3) = ML(i1)*ML(i2)*ML(i3)*MR(j1)*MR(j2)*MR(j3)
  36         Function ML,MR,L,R;
  37         Function KKL,KKR;
  38
  39         Autodeclare Function Jpt;
  40
  41 #procedure DeconcatenateN(F,n)
  42 *
  43 * Deconcatenate function F into n-partitions or all partitions when n==0
  44 *
  45 * n==0: F(1,2,3) --> F(1)*F(2)*F(3) + F(1,2)*F(3) + F(1)*F(2,3)
  46 * n==2: F(1,2,3) --> F(1,2)*F(3) + F(1)*F(2,3)
  47 *
  48         if (match(tmpF1(?m)));
  49                 print "Error: temp function already present: %t";
  50                 exit;
  51         endif;
  52
  53         if ((count('F',1) <= -1) || (count('F',1) >= 2));
  54                 print "Error: Deconc does not work for : %t";
  55                 exit;
  56         endif;
  57
  58         chainout 'F';
  59
  60         repeat id once 'F'(?m)*'F'(?n) =
  61         + 'F'(?m,?n)
  62         + tmpF1(?m)*'F'(?n)
  63         ;
  64
  65         id tmpF1(?m) = 'F'(?m);
  66
  67         #if 'n' == 0
  68 * do nothing, keep all partitions
  69         #else
  70 * keep only n-partitions (or up to n-partitions)
  71 *        if ((match('F'(?m))) && (count('F',1) != 'n'));
  72         if ((match('F'(?m))) && (count('F',1) > 'n'));
  73                 discard;
  74         endif;
  75         #endif
  76
  77 * remove trivial partition F(1,2,3) -> F(1,2,3)
  78         if ((count('F',1) == 1));
  79                 discard;
  80         endif;
  81
  82 #endprocedure
  83
  84
  85 #procedure BiDeconcatenate(phi,n)
  86 *
  87 * phi(A,[p],B) --> all deconcatenations of A and B, separated by [c]
  88 *
  89 * s(1,2,3)*phi(1,2,3,[p],4,5,6) = phi(1,[c],2,3,[p],4,[c],5,6) + phi(1,2,[c],3,[p],4,[c],5,6) + ...
  90
  91 * A and B cannot be composed of smaller words, so no [c]s allowed
  92         if (match(phi(?m,[c],?n)));
  93                 print "Bug: cannot deconcatenate multiple slots %t";
  94                 exit;
  95         endif;
  96
  97         id once 'phi'(?m1,[p],?m2) = Is(?m1)*ML(?m1)*MR(?m2);
  98         #call DeconcatenateN(ML,'n')
  99         #call DeconcatenateN(MR,'n')
 100
 101 * keep only the same depth of deconcatenation on both sides
 102         if (count(ML,1) != count(MR,1));
 103         discard;
 104         endif;
 105
 106 * Kill deconcatenations where there is no matching number of slots
 107 * between ML() and MR(). Recall that ML() gives rise to the A-slots
 108 * in the various BGphiN.h files and MR() the B-slots. These slots
 109 * refer to the definition (3.13) of arXiv:1609.07078v1, namely
 110 * T_{A1 A2...}^{B1 B2...}. If |Ai| != |Bj| for all i,j it means
 111 * that all resulting factors phi(Ai|Bj) vanish by the initial
 112 * condition from eq.(3.3).
 113         id ML(?m) = ML(?m)*countL(nargs_(?m));
 114         id MR(?m) = MR(?m)*countR(nargs_(?m));
 115         id countL(i?)*countR(i?) = 1;
 116 * if any countL() remains, it means there was no matching countR(),
 117 * so kill the term
 118         id countL(i?) = 0;
 119
 120         .sort:bideconc'n';
 121 #endprocedure
 122
 123
 124 #procedure tLalphaExpand(n,w)
 125
 126         #if ({'$Npts'-1-'n'} >= 9 && 'w' >= 7)
 127         #include- BGphiN/BGphi9-Jreg.h
 128         bracket L,R;
 129         .sort:brack9;
 130         keep brackets;
 131         id L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*L(?a7)*L(?a8)*L(?a9)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7)*R(?b8)*R(?b9) =
 132         + phi(?a9,[p],?b9)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*L(?a7)*L(?a8)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7)*R(?b8)
 133         - phi(?a9,[p],?b1)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*L(?a7)*L(?a8)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7)*R(?b8)*R(?b9)
 134         ;
 135         #call MultiparticleConstraint(9)
 136         #endif
 137
 138         #if ({'$Npts'-1-'n'} >= 8 && 'w' >= 6)
 139         #include- BGphiN/BGphi8-Jreg.h
 140         bracket L,R;
 141         .sort:brack8;
 142         keep brackets;
 143         id L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*L(?a7)*L(?a8)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7)*R(?b8) =
 144         + phi(?a8,[p],?b8)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*L(?a7)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7)
 145         - phi(?a8,[p],?b1)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*L(?a7)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7)*R(?b8)
 146         ;
 147         #call MultiparticleConstraint(8)
 148         #endif
 149
 150         #if ({'$Npts'-1-'n'} >= 7 && 'w' >= 5)
 151         #include- BGphiN/BGphi7-Jreg.h
 152         bracket L,R;
 153         .sort:brack7;
 154         keep brackets;
 155         id L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*L(?a7)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7) =
 156         + phi(?a7,[p],?b7)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)
 157         - phi(?a7,[p],?b1)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)*R(?b7)
 158         ;
 159         #call MultiparticleConstraint(7)
 160         #endif
 161
 162         #if ({'$Npts'-1-'n'} >= 6 && 'w' >= 4)
 163         #include- BGphiN/BGphi6-Jreg.h
 164         bracket L,R;
 165         .sort:brack6;
 166         keep brackets;
 167         id L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*L(?a6)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6) =
 168         + phi(?a6,[p],?b6)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5)
 169         - phi(?a6,[p],?b1)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*R(?b2)*R(?b3)*R(?b4)*R(?b5)*R(?b6)
 170         ;
 171         #call MultiparticleConstraint(6)
 172         #endif
 173
 174         #if ({'$Npts'-1-'n'} >= 5 && 'w' >= 3)
 175         #include- BGphiN/BGphi5-Jreg.h
 176         bracket L,R;
 177         .sort:brack5;
 178         keep brackets;
 179         id L(?a1)*L(?a2)*L(?a3)*L(?a4)*L(?a5)*R(?b1)*R(?b2)*R(?b3)*R(?b4)*R(?b5) =
 180         + phi(?a5,[p],?b5)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*R(?b1)*R(?b2)*R(?b3)*R(?b4)
 181         - phi(?a5,[p],?b1)*L(?a1)*L(?a2)*L(?a3)*L(?a4)*R(?b2)*R(?b3)*R(?b4)*R(?b5)
 182         ;
 183         #call MultiparticleConstraint(5)
 184         #endif
 185
 186         #if {'$Npts'-1-'n'} >= 4
 187         #include- BGphiN/BGphi4-Jreg.h
 188         bracket L,R;
 189         .sort:brack4;
 190         keep brackets;
 191         id L(?a1)*L(?a2)*L(?a3)*L(?a4)*R(?b1)*R(?b2)*R(?b3)*R(?b4) =
 192         + phi(?a4,[p],?b4)*L(?a1)*L(?a2)*L(?a3)*R(?b1)*R(?b2)*R(?b3)
 193         - phi(?a4,[p],?b1)*L(?a1)*L(?a2)*L(?a3)*R(?b2)*R(?b3)*R(?b4)
 194         ;
 195         #call MultiparticleConstraint(4)
 196         #endif
 197
 198         #if {'$Npts'-1-'n'} >= 3
 199         #include- BGphiN/BGphi3-Jreg.h
 200         bracket L,R;
 201         .sort:brack3;
 202         keep brackets;
 203         id L(?a1)*L(?a2)*L(?a3)*R(?b1)*R(?b2)*R(?b3) =
 204         + phi(?a3,[p],?b3)*L(?a1)*L(?a2)*R(?b1)*R(?b2)
 205         - phi(?a3,[p],?b1)*L(?a1)*L(?a2)*R(?b2)*R(?b3)
 206         ;
 207         #call MultiparticleConstraint(3)
 208         #endif
 209
 210         bracket L,R;
 211         .sort:brack2;
 212         keep brackets;
 213         id L(?a1)*L(?a2)*R(?b1)*R(?b2) =
 214         + phi(?a2,[p],?b2)*phi(?a1,[p],?b1)
 215         - phi(?a2,[p],?b1)*phi(?a1,[p],?b2)
 216         ;
 217         #call MultiparticleConstraint(2)
 218
 219         id ML(?m1)*ML(?m2)*MR(?n1)*MR(?n2) =
 220         + phi(?m1,[p],?n1)*phi(?m2,[p],?n2)
 221         - phi(?m2,[p],?n1)*phi(?m1,[p],?n2)
 222         ;
 223
 224         bracket phi;
 225         .sort:brackphi;
 226         keep brackets;
 227
 228         #call MultiparticleConstraint(1)
 229
 230 #endprocedure
 231
 232
 233 #procedure MultiparticleConstraint(n)
 234
 235         if (match(phi(?m,[c],?n)));
 236         print "bug: no subwords in phi %t";
 237         exit;
 238         endif;
 239
 240         id phi(i?,[p],j?) = delta_(i,j);
 241         id phi(i?,j?,[p],i?,j?) = Is(i,j);
 242         id phi(i?,j?,[p],j?,i?) = - Is(i,j);
 243
 244 * remove phi(A|B) where length(A) != length(B)
 245         id phi(?m,[p],?n) = phi(?m,[p],?n)*delta_(nargs_(?m),nargs_(?n));
 246
 247 * remove phi(A|B) when letters in A and B are not the same
 248         if (match(ctmpF1(?m))==0);
 249         id phi(?m,[p],?n) = phi(?m,[p],?n)*ctmpF1(?m,[p],?n);
 250         repeat id ctmpF1(?m1,i?,?m2,[p],?m3,i?,?m4) = ctmpF1(?m1,?m2,[p],?m3,?m4);
 251         id ctmpF1([p]) = 1;
 252         id ctmpF1(?m) = 0;
 253         endif;
 254
 255         .sort: MulCtrnt'n';
 256
 257 #endprocedure
 258
 259 #procedure BGexpand(w)
 260
 261         .sort:AlphapWeight'w';
 262         Symbol zwgt(:'w');
 263
 264         #if 'w' > 7
 265         print "Error: order higher than alpha^7 not implemented!";
 266         exit;
 267         #endif
 268
 269 * Sanity check: number of arguments on both sides must be equal
 270         id Zint(?m,[p],?n) = Zint(?m,[p],?n)*nargs(nargs_(?m) - nargs_(?n));
 271         id nargs(0) = 1;
 272         if (match(nargs(i?)));
 273         print "Error: unequal number of labels in domain and integrand: %t";
 274         exit;
 275         endif;
 276
 277         if (match(Zint(?m,[p],?n)) == 0);
 278         print "Error: specify the domain P and integrand Q as Zint(P,[p],Q)";
 279         exit;
 280         endif;
 281
 282 * Get the number of points
 283         id once Zint(?m,[p],?n) = Zint(?m,[p],?n)*nargs(nargs_(?n));
 284         id once nargs(x1?$Npts) = 1;
 285
 286         #define Npts "$Npts"
 287
 288 * Exploit cyclic symmetry of Zint(P|Q) in Q to match last label
 289         id once Zint(?m,i?,[p],?n,i?,?p) = s(?n,?p)*phi(?m,[p],?p,?n);
 290         .sort
 291
 292 * TODO: the number of calls to BiDeconcatenate(phi,i) should vary according to w
 293         #call BiDeconcatenate(phi,{'w'+2})
 294         #call CancelMandelstams('Npts')
 295
 296         #call tLalphaExpand(0,'w')
 297         #call BiDeconcatenate(phi,{'w'+2})
 298         #call tLalphaExpand(1,'w')
 299         #call BiDeconcatenate(phi,{'w'+2})
 300         #call tLalphaExpand(2,'w')
 301         #call BiDeconcatenate(phi,{'w'+2})
 302         #call tLalphaExpand(3,'w')
 303         #call BiDeconcatenate(phi,{'w'+2})
 304         #call tLalphaExpand(4,'w')
 305         #call BiDeconcatenate(phi,{'w'+2})
 306         #call tLalphaExpand(5,'w')
 307         #call BiDeconcatenate(phi,{'w'+2})
 308         #call tLalphaExpand(6,'w')
 309         #call BiDeconcatenate(phi,{'w'+2})
 310         #call tLalphaExpand(7,'w')
 311
 312         if (match(phi(?m)));
 313         print "Bug: cannot have phi() at the end of recursion: %t";
 314         print "Add more calls to tLalphaExpand()";
 315         exit;
 316         endif;
 317
 318         #if ('w' >= 7 && '$Npts' >= 10)
 319         abracket phi,Is,KK,zeta2,zeta3,zeta5,zeta7;
 320         .sort:abrack10;
 321         keep brackets;
 322         #include- NptJregs/10ptJregs.h
 323         #endif
 324
 325         #if ('w' >= 6 && '$Npts' >= 9)
 326         abracket phi,Is,KK,zeta2,zeta3,zeta5,zeta7;
 327         .sort:abrack9;
 328         keep brackets;
 329         #include- NptJregs/9ptJregs.h
 330         #endif
 331
 332         #if ('w' >= 5 && '$Npts' >= 8)
 333         abracket phi,Is,KK,zeta2,zeta3,zeta5,zeta7;
 334         .sort:abrack8;
 335         keep brackets;
 336         #include- NptJregs/8ptJregs.h
 337         #endif
 338
 339         #if ('w' >= 4 && '$Npts' >= 7)
 340         abracket phi,Is,KK,zeta2,zeta3,zeta5,zeta7;
 341         .sort:abrack7;
 342         keep brackets;
 343         #include- NptJregs/7ptJregs.h
 344         #endif
 345
 346         #if ('w' >= 3 && '$Npts' >= 6)
 347         abracket phi,Is,KK,zeta2,zeta3,zeta5,zeta7;
 348         .sort:abrack6;
 349         keep brackets;
 350         #include- NptJregs/6ptJregs.h
 351         #endif
 352
 353         #if ('w' >= 2 && '$Npts' >= 5)
 354         abracket phi,Is,KK,zeta2,zeta3,zeta5,zeta7;
 355         .sort:abrack5;
 356         keep brackets;
 357         #include- NptJregs/5ptJregs.h
 358         #endif
 359
 360         #if ('w' >= 1 && '$Npts' >= 4)
 361         abracket phi,Is,KK,zeta2,zeta3,zeta5,zeta7;
 362         .sort:abrack4;
 363         keep brackets;
 364         #include- NptJregs/4ptJregs.h
 365         #endif
 366
 367         id zwgt = 1;
 368         #call KKTos()
 369
 370 #endprocedure
 371
 372 #procedure KKTos()
 373
 374         bracket KK;
 375         .sort:bracKK;
 376         keep brackets;
 377
 378         repeat id KK(?n,i?,j?,[c],?m) = KK(?n,i,[c],?m) + KK(j,[c],?m);
 379
 380         id KK(i?,[c],i1?) = s(i,i1);
 381         id KK(i?,[c],i1?,i2?) = s(i,i1) + s(i,i2);
 382         id KK(i?,[c],i1?,i2?,i3?) = s(i,i1) + s(i,i2) + s(i,i3);
 383         id KK(i?,[c],i1?,i2?,i3?,i4?) = s(i,i1) + s(i,i2) + s(i,i3) + s(i,i4);
 384         id KK(i?,[c],i1?,i2?,i3?,i4?,i5?) = s(i,i1) + s(i,i2) + s(i,i3) + s(i,i4) + s(i,i5);
 385         id KK(i?,[c],i1?,...,i6?) = s(i,i1) + s(i,i2) + s(i,i3) + s(i,i4) + s(i,i5) + s(i,i6);
 386         id KK(i?,[c],i1?,...,i7?) = s(i,i1) + s(i,i2) + s(i,i3) + s(i,i4) + s(i,i5) + s(i,i6) + s(i,i7);
 387         id KK(i?,[c],i1?,...,i8?) = s(i,i1) + s(i,i2) + s(i,i3) + s(i,i4) + s(i,i5) + s(i,i6) + s(i,i7) + s(i,i8);
 388
 389         .sort:KKTos;
 390 *        #call KNBasis('$Npts')
 391 *        #call MomentumConservationMandelstams('$Npts')
 392 *        #call NumericMandelstams('$Npts')
 393
 394 #endprocedure
 395
 396
 397 #ifndef 'pss2mandelstams'
 398
 399 #procedure CancelMandelstams(N)
 400
 401         #do i=2,'N'
 402         id s(i1?,...,i'i'?)^-1 = Is(i1,...,i'i');
 403         id s(i1?,...,i'i'?)*Is(i1?,...,i'i'?) = 1;
 404         #enddo
 405
 406 #endprocedure
 407
 408 #endif