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) 1999-2003 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 <iostream>
  24 #include <stdexcept>
  25
  26 #include "color.h"
  27 #include "idx.h"
  28 #include "ncmul.h"
  29 #include "symmetry.h"
  30 #include "numeric.h"
  31 #include "mul.h"
  32 #include "power.h" // for sqrt()
  33 #include "symbol.h"
  34 #include "print.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
  45
  46 //////////
  47 // default ctor, dtor, copy ctor, assignment operator and helpers
  48 //////////
  49
  50 color::color() : representation_label(0)
  51 {
  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         tinfo_key = TINFO_color;
  77 }
  78
  79 /** Construct object with one color index. This constructor is for internal
  80  *  use only. Use the color_T() function instead.
  81  *  @see color_T */
  82 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  83 {
  84         tinfo_key = TINFO_color;
  85 }
  86
  87 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  88 {
  89         tinfo_key = TINFO_color;
  90 }
  91
  92 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  93 {
  94         tinfo_key = TINFO_color;
  95 }
  96
  97 //////////
  98 // archiving
  99 //////////
 100
 101 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 102 {
 103         unsigned rl;
 104         n.find_unsigned("label", rl);
 105         representation_label = rl;
 106 }
 107
 108 void color::archive(archive_node &n) const
 109 {
 110         inherited::archive(n);
 111         n.add_unsigned("label", representation_label);
 112 }
 113
 114 DEFAULT_UNARCHIVE(color)
 115 DEFAULT_ARCHIVING(su3one)
 116 DEFAULT_ARCHIVING(su3t)
 117 DEFAULT_ARCHIVING(su3f)
 118 DEFAULT_ARCHIVING(su3d)
 119
 120 //////////
 121 // functions overriding virtual functions from base classes
 122 //////////
 123
 124 int color::compare_same_type(const basic & other) const
 125 {
 126         GINAC_ASSERT(is_a<color>(other));
 127         const color &o = static_cast<const color &>(other);
 128
 129         if (representation_label != o.representation_label) {
 130                 // different representation label
 131                 return representation_label < o.representation_label ? -1 : 1;
 132         }
 133
 134         return inherited::compare_same_type(other);
 135 }
 136
 137 bool color::match_same_type(const basic & other) const
 138 {
 139         GINAC_ASSERT(is_a<color>(other));
 140         const color &o = static_cast<const color &>(other);
 141
 142         return representation_label == o.representation_label;
 143 }
 144
 145 DEFAULT_COMPARE(su3one)
 146 DEFAULT_COMPARE(su3t)
 147 DEFAULT_COMPARE(su3f)
 148 DEFAULT_COMPARE(su3d)
 149
 150 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 151 DEFAULT_PRINT(su3t, "T")
 152 DEFAULT_PRINT(su3f, "f")
 153 DEFAULT_PRINT(su3d, "d")
 154
 155 /** Perform automatic simplification on noncommutative product of color
 156  *  objects. This removes superfluous ONEs. */
 157 ex color::simplify_ncmul(const exvector & v) const
 158 {
 159         exvector s;
 160         s.reserve(v.size());
 161
 162         // Remove superfluous ONEs
 163         exvector::const_iterator it = v.begin(), itend = v.end();
 164         while (it != itend) {
 165                 if (!is_ex_of_type(it->op(0), su3one))
 166                         s.push_back(*it);
 167                 it++;
 168         }
 169
 170         if (s.empty())
 171                 return color(su3one(), representation_label);
 172         else
 173                 return simplified_ncmul(s);
 174 }
 175
 176 ex color::thisexprseq(const exvector & v) const
 177 {
 178         return color(representation_label, v);
 179 }
 180
 181 ex color::thisexprseq(exvector * vp) const
 182 {
 183         return color(representation_label, vp);
 184 }
 185
 186 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 187  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 188  *  iv2 and the sign introduced by permuting that index to the front.
 189  *
 190  *  @param iv3 Vector of 3 indices
 191  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 192  *  @param sig Returs sign introduced by index permutation
 193  *  @return the free index (the one that is in iv3 but not in iv2) */
 194 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 195 {
 196         GINAC_ASSERT(iv3.size() == 3);
 197         GINAC_ASSERT(iv2.size() == 2);
 198
 199         sig = 1;
 200
 201 #define TEST_PERMUTATION(A,B,C,P) \
 202         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 203                 sig = P; \
 204                 return iv3[A]; \
 205         }
 206
 207         TEST_PERMUTATION(0,1,2,  1);
 208         TEST_PERMUTATION(0,2,1, -1);
 209         TEST_PERMUTATION(1,0,2, -1);
 210         TEST_PERMUTATION(1,2,0,  1);
 211         TEST_PERMUTATION(2,0,1,  1);
 212         TEST_PERMUTATION(2,1,0, -1);
 213
 214         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 215 }
 216
 217 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 218 ex su3d::eval_indexed(const basic & i) const
 219 {
 220         GINAC_ASSERT(is_a<indexed>(i));
 221         GINAC_ASSERT(i.nops() == 4);
 222         GINAC_ASSERT(is_a<su3d>(i.op(0)));
 223
 224         // Convolutions are zero
 225         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 226                 return _ex0;
 227
 228         // Numeric evaluation
 229         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 230
 231                 // Sort indices
 232                 int v[3];
 233                 for (unsigned j=0; j<3; j++)
 234                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 235                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 236                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 237                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 238
 239 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 240
 241                 // Check for non-zero elements
 242                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 243                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 244                         return _ex1_2;
 245                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 246                         return _ex_1_2;
 247                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 248                         return sqrt(_ex3)*_ex1_3;
 249                 else if (CMPINDICES(8,8,8))
 250                         return sqrt(_ex3)*_ex_1_3;
 251                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 252                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 253                         return sqrt(_ex3)/_ex_6;
 254                 else
 255                         return _ex0;
 256         }
 257
 258         // No further simplifications
 259         return i.hold();
 260 }
 261
 262 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 263 ex su3f::eval_indexed(const basic & i) const
 264 {
 265         GINAC_ASSERT(is_a<indexed>(i));
 266         GINAC_ASSERT(i.nops() == 4);
 267         GINAC_ASSERT(is_a<su3f>(i.op(0)));
 268
 269         // Numeric evaluation
 270         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 271
 272                 // Sort indices, remember permutation sign
 273                 int v[3];
 274                 for (unsigned j=0; j<3; j++)
 275                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 276                 int sign = 1;
 277                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 278                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 279                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 280
 281                 // Check for non-zero elements
 282                 if (CMPINDICES(1,2,3))
 283                         return sign;
 284                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 285                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 286                         return _ex1_2 * sign;
 287                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 288                         return _ex_1_2 * sign;
 289                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 290                         return sqrt(_ex3)/2 * sign;
 291                 else
 292                         return _ex0;
 293         }
 294
 295         // No further simplifications
 296         return i.hold();
 297 }
 298
 299
 300 /** Contraction of generator with something else. */
 301 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 302 {
 303         GINAC_ASSERT(is_a<indexed>(*self));
 304         GINAC_ASSERT(is_a<indexed>(*other));
 305         GINAC_ASSERT(self->nops() == 2);
 306         GINAC_ASSERT(is_a<su3t>(self->op(0)));
 307         unsigned char rl = ex_to<color>(*self).get_representation_label();
 308
 309         if (is_ex_exactly_of_type(other->op(0), su3t)) {
 310
 311                 // Contraction only makes sense if the represenation labels are equal
 312                 GINAC_ASSERT(is_a<color>(*other));
 313                 if (ex_to<color>(*other).get_representation_label() != rl)
 314                         return false;
 315
 316                 // T.a T.a = 4/3 ONE
 317                 if (other - self == 1) {
 318                         *self = numeric(4, 3);
 319                         *other = color_ONE(rl);
 320                         return true;
 321
 322                 // T.a T.b T.a = -1/6 T.b
 323                 } else if (other - self == 2
 324                         && is_ex_of_type(self[1], color)) {
 325                         *self = numeric(-1, 6);
 326                         *other = _ex1;
 327                         return true;
 328
 329                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 330                 } else {
 331                         exvector::iterator it = self + 1;
 332                         while (it != other) {
 333                                 if (!is_ex_of_type(*it, color)) {
 334                                         return false;
 335                                 }
 336                                 it++;
 337                         }
 338
 339                         it = self + 1;
 340                         ex S = _ex1;
 341                         while (it != other) {
 342                                 S *= *it;
 343                                 *it++ = _ex1;
 344                         }
 345
 346                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 347                         *other = _ex1;
 348                         return true;
 349                 }
 350         }
 351
 352         return false;
 353 }
 354
 355 /** Contraction of an indexed symmetric structure constant with something else. */
 356 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 357 {
 358         GINAC_ASSERT(is_a<indexed>(*self));
 359         GINAC_ASSERT(is_a<indexed>(*other));
 360         GINAC_ASSERT(self->nops() == 4);
 361         GINAC_ASSERT(is_a<su3d>(self->op(0)));
 362
 363         if (is_ex_exactly_of_type(other->op(0), su3d)) {
 364
 365                 // Find the dummy indices of the contraction
 366                 exvector self_indices = ex_to<indexed>(*self).get_indices();
 367                 exvector other_indices = ex_to<indexed>(*other).get_indices();
 368                 exvector all_indices = self_indices;
 369                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 370                 exvector free_indices, dummy_indices;
 371                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 372
 373                 // d.abc d.abc = 40/3
 374                 if (dummy_indices.size() == 3) {
 375                         *self = numeric(40, 3);
 376                         *other = _ex1;
 377                         return true;
 378
 379                 // d.akl d.bkl = 5/3 delta.ab
 380                 } else if (dummy_indices.size() == 2) {
 381                         exvector a;
 382                         std::back_insert_iterator<exvector> ita(a);
 383                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 384                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 385                         GINAC_ASSERT(a.size() == 2);
 386                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 387                         *other = _ex1;
 388                         return true;
 389                 }
 390
 391         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 392
 393                 // d.abc T.b T.c = 5/6 T.a
 394                 if (other+1 != v.end()
 395                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 396                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 397
 398                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 399                         exvector dummy_indices;
 400                         dummy_indices.push_back(other[0].op(1));
 401                         dummy_indices.push_back(other[1].op(1));
 402                         int sig;
 403                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 404                         *self = numeric(5, 6);
 405                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 406                         other[1] = _ex1;
 407                         return true;
 408                 }
 409         }
 410
 411         return false;
 412 }
 413
 414 /** Contraction of an indexed antisymmetric structure constant with something else. */
 415 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 416 {
 417         GINAC_ASSERT(is_a<indexed>(*self));
 418         GINAC_ASSERT(is_a<indexed>(*other));
 419         GINAC_ASSERT(self->nops() == 4);
 420         GINAC_ASSERT(is_a<su3f>(self->op(0)));
 421
 422         if (is_ex_exactly_of_type(other->op(0), su3f)) { // f*d is handled by su3d class
 423
 424                 // Find the dummy indices of the contraction
 425                 exvector dummy_indices;
 426                 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
 427
 428                 // f.abc f.abc = 24
 429                 if (dummy_indices.size() == 3) {
 430                         *self = 24;
 431                         *other = _ex1;
 432                         return true;
 433
 434                 // f.akl f.bkl = 3 delta.ab
 435                 } else if (dummy_indices.size() == 2) {
 436                         int sign1, sign2;
 437                         ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
 438                         ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
 439                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 440                         *other = _ex1;
 441                         return true;
 442                 }
 443
 444         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 445
 446                 // f.abc T.b T.c = 3/2 I T.a
 447                 if (other+1 != v.end()
 448                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 449                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 450
 451                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 452                         exvector dummy_indices;
 453                         dummy_indices.push_back(other[0].op(1));
 454                         dummy_indices.push_back(other[1].op(1));
 455                         int sig;
 456                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 457                         *self = numeric(3, 2) * sig * I;
 458                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 459                         other[1] = _ex1;
 460                         return true;
 461                 }
 462         }
 463
 464         return false;
 465 }
 466
 467 //////////
 468 // global functions
 469 //////////
 470
 471 ex color_ONE(unsigned char rl)
 472 {
 473         return color(su3one(), rl);
 474 }
 475
 476 ex color_T(const ex & a, unsigned char rl)
 477 {
 478         if (!is_ex_of_type(a, idx))
 479                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 480         if (!ex_to<idx>(a).get_dim().is_equal(8))
 481                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 482
 483         return color(su3t(), a, rl);
 484 }
 485
 486 ex color_f(const ex & a, const ex & b, const ex & c)
 487 {
 488         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 489                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 490         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))
 491                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 492
 493         return indexed(su3f(), sy_anti(), a, b, c);
 494 }
 495
 496 ex color_d(const ex & a, const ex & b, const ex & c)
 497 {
 498         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 499                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 500         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))
 501                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 502
 503         return indexed(su3d(), sy_symm(), a, b, c);
 504 }
 505
 506 ex color_h(const ex & a, const ex & b, const ex & c)
 507 {
 508         return color_d(a, b, c) + I * color_f(a, b, c);
 509 }
 510
 511 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 512  *  is that of a color object with the specified representation label. */
 513 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 514 {
 515         return ti == (TINFO_color + rl);
 516 }
 517
 518 ex color_trace(const ex & e, unsigned char rl)
 519 {
 520         if (is_ex_of_type(e, color)) {
 521
 522                 if (ex_to<color>(e).get_representation_label() == rl
 523                  && is_ex_of_type(e.op(0), su3one))
 524                         return _ex3;
 525                 else
 526                         return _ex0;
 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 (unsigned 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         } else if (e.nops() > 0) {
 586
 587                 // Trace maps to all other container classes (this includes sums)
 588                 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);
 589                 return e.map(fcn);
 590
 591         } else
 592                 return _ex0;
 593 }
 594
 595 } // namespace GiNaC