ginac/color.cpp

   1 /** @file color.cpp
   2  *
   3  *  Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <algorithm>
  24 #include <stdexcept>
  25
  26 #include "color.h"
  27 #include "idx.h"
  28 #include "ncmul.h"
  29 #include "numeric.h"
  30 #include "power.h" // for sqrt()
  31 #include "symbol.h"
  32 #include "print.h"
  33 #include "archive.h"
  34 #include "debugmsg.h"
  35 #include "utils.h"
  36
  37 namespace GiNaC {
  38
  39 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
  44
  45 //////////
  46 // default constructor, destructor, copy constructor assignment operator and helpers
  47 //////////
  48
  49 color::color() : representation_label(0)
  50 {
  51         debugmsg("color default constructor", LOGLEVEL_CONSTRUCT);
  52         tinfo_key = TINFO_color;
  53 }
  54
  55 void color::copy(const color & other)
  56 {
  57         inherited::copy(other);
  58         representation_label = other.representation_label;
  59 }
  60
  61 DEFAULT_DESTROY(color)
  62 DEFAULT_CTORS(su3one)
  63 DEFAULT_CTORS(su3t)
  64 DEFAULT_CTORS(su3f)
  65 DEFAULT_CTORS(su3d)
  66
  67 //////////
  68 // other constructors
  69 //////////
  70
  71 /** Construct object without any color index. This constructor is for
  72  *  internal use only. Use the color_ONE() function instead.
  73  *  @see color_ONE */
  74 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  75 {
  76         debugmsg("color constructor from ex,unsigned char", LOGLEVEL_CONSTRUCT);
  77         tinfo_key = TINFO_color;
  78 }
  79
  80 /** Construct object with one color index. This constructor is for internal
  81  *  use only. Use the color_T() function instead.
  82  *  @see color_T */
  83 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  84 {
  85         debugmsg("color constructor from ex,ex,unsigned char", LOGLEVEL_CONSTRUCT);
  86         tinfo_key = TINFO_color;
  87 }
  88
  89 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
  90 {
  91         debugmsg("color constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  92         tinfo_key = TINFO_color;
  93 }
  94
  95 color::color(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
  96 {
  97         debugmsg("color constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  98         tinfo_key = TINFO_color;
  99 }
 100
 101 //////////
 102 // archiving
 103 //////////
 104
 105 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 106 {
 107         debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
 108         unsigned rl;
 109         n.find_unsigned("label", rl);
 110         representation_label = rl;
 111 }
 112
 113 void color::archive(archive_node &n) const
 114 {
 115         inherited::archive(n);
 116         n.add_unsigned("label", representation_label);
 117 }
 118
 119 DEFAULT_UNARCHIVE(color)
 120 DEFAULT_ARCHIVING(su3one)
 121 DEFAULT_ARCHIVING(su3t)
 122 DEFAULT_ARCHIVING(su3f)
 123 DEFAULT_ARCHIVING(su3d)
 124
 125 //////////
 126 // functions overriding virtual functions from bases classes
 127 //////////
 128
 129 int color::compare_same_type(const basic & other) const
 130 {
 131         GINAC_ASSERT(other.tinfo() == TINFO_color);
 132         const color &o = static_cast<const color &>(other);
 133
 134         if (representation_label != o.representation_label) {
 135                 // different representation label
 136                 return representation_label < o.representation_label ? -1 : 1;
 137         }
 138
 139         return inherited::compare_same_type(other);
 140 }
 141
 142 DEFAULT_COMPARE(su3one)
 143 DEFAULT_COMPARE(su3t)
 144 DEFAULT_COMPARE(su3f)
 145 DEFAULT_COMPARE(su3d)
 146
 147 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 148 DEFAULT_PRINT(su3t, "T")
 149 DEFAULT_PRINT(su3f, "f")
 150 DEFAULT_PRINT(su3d, "d")
 151
 152 /** Perform automatic simplification on noncommutative product of color
 153  *  objects. This removes superfluous ONEs. */
 154 ex color::simplify_ncmul(const exvector & v) const
 155 {
 156         exvector s;
 157         s.reserve(v.size());
 158
 159         // Remove superfluous ONEs
 160         exvector::const_iterator it = v.begin(), itend = v.end();
 161         while (it != itend) {
 162                 if (!is_ex_of_type(it->op(0), su3one))
 163                         s.push_back(*it);
 164                 it++;
 165         }
 166
 167         if (s.size() == 0)
 168                 return color(su3one(), representation_label);
 169         else
 170                 return simplified_ncmul(s);
 171 }
 172
 173 ex color::thisexprseq(const exvector & v) const
 174 {
 175         return color(representation_label, v);
 176 }
 177
 178 ex color::thisexprseq(exvector * vp) const
 179 {
 180         return color(representation_label, vp);
 181 }
 182
 183 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 184  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 185  *  iv2 and the sign introduced by permuting that index to the front.
 186  *
 187  *  @param iv3 Vector of 3 indices
 188  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 189  *  @param sig Returs sign introduced by index permutation
 190  *  @return the free index (the one that is in iv3 but not in iv2) */
 191 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 192 {
 193         GINAC_ASSERT(iv3.size() == 3);
 194         GINAC_ASSERT(iv2.size() == 2);
 195
 196         sig = 1;
 197
 198 #define TEST_PERMUTATION(A,B,C,P) \
 199         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 200                 sig = P; \
 201                 return iv3[A]; \
 202         }
 203
 204         TEST_PERMUTATION(0,1,2,  1);
 205         TEST_PERMUTATION(0,2,1, -1);
 206         TEST_PERMUTATION(1,0,2, -1);
 207         TEST_PERMUTATION(1,2,0,  1);
 208         TEST_PERMUTATION(2,0,1,  1);
 209         TEST_PERMUTATION(2,1,0, -1);
 210
 211         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 212 }
 213
 214 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 215 ex su3d::eval_indexed(const basic & i) const
 216 {
 217         GINAC_ASSERT(is_of_type(i, indexed));
 218         GINAC_ASSERT(i.nops() == 4);
 219         GINAC_ASSERT(is_ex_of_type(i.op(0), su3d));
 220
 221         // Convolutions are zero
 222         if (static_cast<const indexed &>(i).get_dummy_indices().size() != 0)
 223                 return _ex0();
 224
 225         // Numeric evaluation
 226         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 227
 228                 // Sort indices
 229                 int v[3];
 230                 for (unsigned j=0; j<3; j++)
 231                         v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
 232                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 233                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 234                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 235
 236 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 237
 238                 // Check for non-zero elements
 239                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 240                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 241                         return _ex1_2();
 242                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 243                         return _ex_1_2();
 244                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 245                         return sqrt(_ex3())/3;
 246                 else if (CMPINDICES(8,8,8))
 247                         return -sqrt(_ex3())/3;
 248                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 249                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 250                         return -sqrt(_ex3())/6;
 251                 else
 252                         return _ex0();
 253         }
 254
 255         // No further simplifications
 256         return i.hold();
 257 }
 258
 259 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 260 ex su3f::eval_indexed(const basic & i) const
 261 {
 262         GINAC_ASSERT(is_of_type(i, indexed));
 263         GINAC_ASSERT(i.nops() == 4);
 264         GINAC_ASSERT(is_ex_of_type(i.op(0), su3f));
 265
 266         // Numeric evaluation
 267         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 268
 269                 // Sort indices, remember permutation sign
 270                 int v[3];
 271                 for (unsigned j=0; j<3; j++)
 272                         v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
 273                 int sign = 1;
 274                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 275                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 276                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 277
 278                 // Check for non-zero elements
 279                 if (CMPINDICES(1,2,3))
 280                         return sign;
 281                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 282                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 283                         return _ex1_2() * sign;
 284                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 285                         return _ex_1_2() * sign;
 286                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 287                         return sqrt(_ex3())/2 * sign;
 288                 else
 289                         return _ex0();
 290         }
 291
 292         // No further simplifications
 293         return i.hold();
 294 }
 295
 296
 297 /** Contraction of generator with something else. */
 298 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 299 {
 300         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 301         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 302         GINAC_ASSERT(self->nops() == 2);
 303         GINAC_ASSERT(is_ex_of_type(self->op(0), su3t));
 304         unsigned char rl = ex_to_color(*self).get_representation_label();
 305
 306         if (is_ex_exactly_of_type(other->op(0), su3t)) {
 307
 308                 // T.a T.a = 4/3 ONE
 309                 if (other - self == 1) {
 310                         *self = numeric(4, 3);
 311                         *other = color_ONE(rl);
 312                         return true;
 313
 314                 // T.a T.b T.a = -1/6 T.b
 315                 } else if (other - self == 2
 316                         && is_ex_of_type(self[1], color)) {
 317                         *self = numeric(-1, 6);
 318                         *other = _ex1();
 319                         return true;
 320
 321                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 322                 } else {
 323                         exvector::iterator it = self + 1;
 324                         while (it != other) {
 325                                 if (!is_ex_of_type(*it, color)) {
 326                                         return false;
 327                                 }
 328                                 it++;
 329                         }
 330
 331                         it = self + 1;
 332                         ex S = _ex1();
 333                         while (it != other) {
 334                                 S *= *it;
 335                                 *it++ = _ex1();
 336                         }
 337
 338                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 339                         *other = _ex1();
 340                         return true;
 341                 }
 342         }
 343
 344         return false;
 345 }
 346
 347 /** Contraction of an indexed symmetric structure constant with something else. */
 348 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 349 {
 350         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 351         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 352         GINAC_ASSERT(self->nops() == 4);
 353         GINAC_ASSERT(is_ex_of_type(self->op(0), su3d));
 354
 355         if (is_ex_exactly_of_type(other->op(0), su3d)) {
 356
 357                 // Find the dummy indices of the contraction
 358                 exvector self_indices = ex_to_indexed(*self).get_indices();
 359                 exvector other_indices = ex_to_indexed(*other).get_indices();
 360                 exvector all_indices = self_indices;
 361                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 362                 exvector free_indices, dummy_indices;
 363                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 364
 365                 // d.abc d.abc = 40/3
 366                 if (dummy_indices.size() == 3) {
 367                         *self = numeric(40, 3);
 368                         *other = _ex1();
 369                         return true;
 370
 371                 // d.akl d.bkl = 5/3 delta.ab
 372                 } else if (dummy_indices.size() == 2) {
 373                         exvector a;
 374                         back_insert_iterator<exvector> ita(a);
 375                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 376                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 377                         GINAC_ASSERT(a.size() == 2);
 378                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 379                         *other = _ex1();
 380                         return true;
 381                 }
 382
 383         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 384
 385                 // d.abc T.b T.c = 5/6 T.a
 386                 if (other+1 != v.end()
 387                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 388                  && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
 389
 390                         exvector self_indices = ex_to_indexed(*self).get_indices();
 391                         exvector dummy_indices;
 392                         dummy_indices.push_back(other[0].op(1));
 393                         dummy_indices.push_back(other[1].op(1));
 394                         int sig;
 395                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 396                         *self = numeric(5, 6);
 397                         other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
 398                         other[1] = _ex1();
 399                         return true;
 400                 }
 401         }
 402
 403         return false;
 404 }
 405
 406 /** Contraction of an indexed antisymmetric structure constant with something else. */
 407 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 408 {
 409         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 410         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 411         GINAC_ASSERT(self->nops() == 4);
 412         GINAC_ASSERT(is_ex_of_type(self->op(0), su3f));
 413
 414         if (is_ex_exactly_of_type(other->op(0), su3f)) { // f*d is handled by su3d class
 415
 416                 // Find the dummy indices of the contraction
 417                 exvector dummy_indices;
 418                 dummy_indices = ex_to_indexed(*self).get_dummy_indices(ex_to_indexed(*other));
 419
 420                 // f.abc f.abc = 24
 421                 if (dummy_indices.size() == 3) {
 422                         *self = 24;
 423                         *other = _ex1();
 424                         return true;
 425
 426                 // f.akl f.bkl = 3 delta.ab
 427                 } else if (dummy_indices.size() == 2) {
 428                         int sign1, sign2;
 429                         ex a = permute_free_index_to_front(ex_to_indexed(*self).get_indices(), dummy_indices, sign1);
 430                         ex b = permute_free_index_to_front(ex_to_indexed(*other).get_indices(), dummy_indices, sign2);
 431                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 432                         *other = _ex1();
 433                         return true;
 434                 }
 435
 436         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 437
 438                 // f.abc T.b T.c = 3/2 I T.a
 439                 if (other+1 != v.end()
 440                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 441                  && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
 442
 443                         exvector self_indices = ex_to_indexed(*self).get_indices();
 444                         exvector dummy_indices;
 445                         dummy_indices.push_back(other[0].op(1));
 446                         dummy_indices.push_back(other[1].op(1));
 447                         int sig;
 448                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 449                         *self = numeric(3, 2) * sig * I;
 450                         other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
 451                         other[1] = _ex1();
 452                         return true;
 453                 }
 454         }
 455
 456         return false;
 457 }
 458
 459 //////////
 460 // global functions
 461 //////////
 462
 463 ex color_ONE(unsigned char rl)
 464 {
 465         return color(su3one(), rl);
 466 }
 467
 468 ex color_T(const ex & a, unsigned char rl)
 469 {
 470         if (!is_ex_of_type(a, idx))
 471                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 472         if (!ex_to_idx(a).get_dim().is_equal(8))
 473                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 474
 475         return color(su3t(), a, rl);
 476 }
 477
 478 ex color_f(const ex & a, const ex & b, const ex & c)
 479 {
 480         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 481                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 482         if (!ex_to_idx(a).get_dim().is_equal(8) || !ex_to_idx(b).get_dim().is_equal(8) || !ex_to_idx(c).get_dim().is_equal(8))
 483                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 484
 485         return indexed(su3f(), indexed::antisymmetric, a, b, c);
 486 }
 487
 488 ex color_d(const ex & a, const ex & b, const ex & c)
 489 {
 490         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 491                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 492         if (!ex_to_idx(a).get_dim().is_equal(8) || !ex_to_idx(b).get_dim().is_equal(8) || !ex_to_idx(c).get_dim().is_equal(8))
 493                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 494
 495         return indexed(su3d(), indexed::symmetric, a, b, c);
 496 }
 497
 498 ex color_h(const ex & a, const ex & b, const ex & c)
 499 {
 500         return color_d(a, b, c) + I * color_f(a, b, c);
 501 }
 502
 503 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 504  *  is that of a color object with the specified representation label. */
 505 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 506 {
 507         return ti == (TINFO_color + rl);
 508 }
 509
 510 ex color_trace(const ex & e, unsigned char rl)
 511 {
 512         if (is_ex_of_type(e, color)) {
 513
 514                 if (ex_to_color(e).get_representation_label() == rl
 515                  && is_ex_of_type(e.op(0), su3one))
 516                         return _ex3();
 517                 else
 518                         return _ex0();
 519
 520         } else if (is_ex_exactly_of_type(e, add)) {
 521
 522                 // Trace of sum = sum of traces
 523                 ex sum = _ex0();
 524                 for (unsigned i=0; i<e.nops(); i++)
 525                         sum += color_trace(e.op(i), rl);
 526                 return sum;
 527
 528         } else if (is_ex_exactly_of_type(e, mul)) {
 529
 530                 // Trace of product: pull out non-color factors
 531                 ex prod = _ex1();
 532                 for (unsigned i=0; i<e.nops(); i++) {
 533                         const ex &o = e.op(i);
 534                         if (is_color_tinfo(o.return_type_tinfo(), rl))
 535                                 prod *= color_trace(o, rl);
 536                         else
 537                                 prod *= o;
 538                 }
 539                 return prod;
 540
 541         } else if (is_ex_exactly_of_type(e, ncmul)) {
 542
 543                 if (!is_color_tinfo(e.return_type_tinfo(), rl))
 544                         return _ex0();
 545
 546                 // Expand product, if necessary
 547                 ex e_expanded = e.expand();
 548                 if (!is_ex_of_type(e_expanded, ncmul))
 549                         return color_trace(e_expanded, rl);
 550
 551                 unsigned num = e.nops();
 552
 553                 if (num == 2) {
 554
 555                         // Tr T_a T_b = 1/2 delta_a_b
 556                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 557
 558                 } else if (num == 3) {
 559
 560                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 561                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 562
 563                 } else {
 564
 565                         // Traces of 4 or more generators are computed recursively:
 566                         // Tr T_a1 .. T_an =
 567                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 568                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 569                         const ex &last_index = e.op(num - 1).op(1);
 570                         const ex &next_to_last_index = e.op(num - 2).op(1);
 571                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 572
 573                         exvector v1;
 574                         v1.reserve(num - 2);
 575                         for (int i=0; i<num-2; i++)
 576                                 v1.push_back(e.op(i));
 577
 578                         exvector v2 = v1;
 579                         v2.push_back(color_T(summation_index, rl));
 580
 581                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 582                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 583                 }
 584         }
 585
 586         return _ex0();
 587 }
 588
 589 } // namespace GiNaC