813:
by using the low-energy effective theory that only contains the massless degrees of freedom by integrating out massive fields. Since it must be again gauge invariant by adding the same five-dimensional Chern–Simons term, the 't Hooft anomaly does not change by integrating out massive degrees of
583:
we have added go to zero, one gets back to the original theory, plus the fermions we have added; the latter remain good degrees of freedom at every energy scale, as they are free fermions at this limit. The gauge symmetry anomaly can be computed at any energy scale, and must always be zero, so that
859:
In the context of quantum field theory, “UV” actually means the high-energy limit of a theory, and “IR” means the low-energy limit, by analogy to the upper and lower peripheries of visible light, but not actually meaning either light or those particular
286:
The anomaly matching condition by G. 't Hooft proposes that a 't Hooft anomaly of continuous symmetry can be computed both in the high-energy and low-energy degrees of freedom (“UV” and “IR”) and give the same answer.
508:
408:
276:
584:
the theory is consistent. One may now get the anomaly of the symmetry in the original theory by subtracting the free fermions we have added, and the result is independent of the energy scale.
35:
for the flavor symmetry must not depend on what scale is chosen for the calculation if it is done by using the degrees of freedom of the theory at some energy scale. It is also known as the
514:
at the far low energy limit (far “IR” ) or the degrees of freedom at the far high energy limit (far “UV”) in order to calculate the anomaly. In the former case one should only consider
592:
Another way to prove the anomaly matching for continuous symmetries is to use the anomaly inflow mechanism. To be specific, we consider four-dimensional spacetime in the following.
145:
811:
711:
662:
172:
106:
771:
751:
731:
682:
633:
613:
74:, it means that the symmetry is exact as a global symmetry of the quantum theory, but there is some impediment to using it as a gauge in the theory.
511:
413:
313:
181:
910:
897:'t Hooft, G. (1980). "Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking". In 't Hooft, G. (ed.).
823:
926:
Kapustin, A.; Thorngren, R. (2014). "Anomalous discrete symmetries in three dimensions and group cohomology".
1111:
529:
of the underlying short-distance theory. In both cases, the answer must be the same. Indeed, in the case of
1106:
1026:
Callan, Jr., C.G.; Harvey, J.A. (1985). "Anomalies and fermion zero modes on strings and domain walls".
538:
522:
525:
which may be composite particles, while in the latter case one should only consider the elementary
778:
530:
296:
111:
78:
787:
687:
638:
1074:
1035:
1000:
945:
150:
84:
52:
20:
8:
1078:
1039:
1004:
949:
510:
becomes anomalous when the background gauge field is introduced. One may use either the
969:
935:
833:
756:
736:
716:
667:
618:
598:
28:
1086:
1047:
1012:
961:
906:
580:
565:
554:
973:
1082:
1043:
1008:
957:
953:
828:
278:, which is often called the flavor symmetry, and this has a 't Hooft anomaly.
51:
There are two closely related but different types of obstructions to formulating a
986:
549:
One proves this condition by the following procedure: we may add to the theory a
534:
753:
and it cannot be restored by adding any four-dimensional local counter terms of
870:
56:
32:
1100:
774:
573:
175:
965:
902:
777:
shows that we can make it gauge invariant by adding the five-dimensional
569:
550:
989:"The axial anomaly and the bound state spectrum in confining theories"
988:
1062:
940:
784:
With the extra dimension, we can now define the effective action
572:, and cancel the anomaly (so that the gauge symmetry will remain
561:
526:
518:
987:
Frishman, Y.; Scwimmer, A.; Banks, T.; Yankielowicz, S. (1981).
503:{\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{V}}
403:{\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{V}}
271:{\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{V}}
558:
307:
515:
281:
925:
537:
breaking occurs and the Wess–Zumino–Witten term for the
557:
to the current related with this symmetry, as well as
77:
As an example of a 't Hooft anomaly, we consider
790:
759:
739:
719:
690:
670:
641:
621:
601:
416:
316:
184:
153:
114:
87:
892:
890:
733:gauge transformation on the background gauge field
805:
765:
745:
725:
705:
676:
656:
627:
607:
502:
402:
270:
166:
139:
100:
1025:
70:If we say that the symmetry of the theory has a '
1098:
887:
873:or instantons so is not included in the example.
41:'t Hooft UV-IR anomaly matching condition
896:
855:
853:
851:
849:
1063:"Consequences of anomalous ward identities"
869:. The axial U(1) symmetry is broken by the
1060:
664:. If there is a 't Hooft anomaly for
615:, we introduce the background gauge field
939:
846:
282:Anomaly matching for continuous symmetry
1099:
310:. This theory has the flavor symmetry
178:. This theory has the global symmetry
899:Recent Developments in Gauge Theories
46:
587:
31:states that the calculation of any
13:
791:
691:
642:
16:Principle in quantum field theory
14:
1123:
775:Wess–Zumino consistency condition
635:and compute the effective action
595:For global continuous symmetries
55:that are both called anomalies:
108:massless fermions: This is the
1054:
1019:
980:
958:10.1103/PhysRevLett.112.231602
919:
863:
800:
794:
700:
694:
651:
645:
576:, as needed for consistency).
491:
484:
469:
455:
437:
423:
391:
384:
369:
355:
337:
323:
259:
252:
237:
223:
205:
191:
134:
121:
1:
1061:Wess, J.; Zumino, B. (1971).
880:
1087:10.1016/0370-2693(71)90582-X
1048:10.1016/0550-3213(85)90489-4
1013:10.1016/0550-3213(81)90268-6
7:
817:
713:is not invariant under the
10:
1128:
824:'t Hooft–Polyakov monopole
295:For example, consider the
290:
25:anomaly matching condition
140:{\displaystyle SU(N_{c})}
839:
544:
541:reproduces the anomaly.
928:Physical Review Letters
806:{\displaystyle \Gamma }
706:{\displaystyle \Gamma }
684:, the effective action
657:{\displaystyle \Gamma }
579:In the limit where the
47:'t Hooft anomalies
37:'t Hooft condition
807:
767:
747:
727:
707:
678:
658:
629:
609:
539:Nambu–Goldstone bosons
523:Nambu–Goldstone bosons
504:
404:
297:quantum chromodynamics
272:
168:
141:
102:
79:quantum chromodynamics
65:t Hooft anomalies
808:
768:
748:
728:
708:
679:
659:
630:
610:
505:
410:This flavor symmetry
405:
273:
169:
167:{\displaystyle N_{f}}
142:
103:
101:{\displaystyle N_{f}}
1112:Quantum field theory
788:
757:
737:
717:
688:
668:
639:
619:
599:
414:
314:
182:
151:
112:
85:
72:t Hooft anomaly
53:quantum field theory
21:quantum field theory
1107:Anomalies (physics)
1079:1971PhLB...37...95W
1040:1985NuPhB.250..427C
1005:1981NuPhB.177..157F
950:2014PhRvL.112w1602K
779:Chern–Simons action
803:
763:
743:
723:
703:
674:
654:
625:
605:
581:coupling constants
512:degrees of freedom
500:
400:
268:
164:
147:gauge theory with
137:
98:
1067:Physics Letters B
1028:Nuclear Physics B
993:Nuclear Physics B
912:978-0-306-40479-5
766:{\displaystyle A}
746:{\displaystyle A}
726:{\displaystyle G}
677:{\displaystyle G}
628:{\displaystyle A}
608:{\displaystyle G}
588:Alternative proof
59:Adler–Bell–Jackiw
1119:
1091:
1090:
1058:
1052:
1051:
1034:(1–4): 427–436.
1023:
1017:
1016:
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978:
977:
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923:
917:
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724:
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683:
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345:
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335:
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277:
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213:
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203:
202:
173:
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162:
146:
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138:
133:
132:
107:
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104:
99:
97:
96:
1127:
1126:
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1120:
1118:
1117:
1116:
1097:
1096:
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1059:
1055:
1024:
1020:
985:
981:
924:
920:
913:
895:
888:
883:
878:
877:
868:
864:
858:
847:
842:
834:'t Hooft symbol
820:
789:
786:
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758:
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754:
738:
735:
734:
718:
715:
714:
689:
686:
685:
669:
666:
665:
640:
637:
636:
620:
617:
616:
600:
597:
596:
590:
547:
535:chiral symmetry
494:
490:
472:
468:
462:
458:
440:
436:
430:
426:
415:
412:
411:
394:
390:
372:
368:
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358:
340:
336:
330:
326:
315:
312:
311:
305:
293:
284:
262:
258:
240:
236:
230:
226:
208:
204:
198:
194:
183:
180:
179:
158:
154:
152:
149:
148:
128:
124:
113:
110:
109:
92:
88:
86:
83:
82:
49:
29:Gerard 't Hooft
17:
12:
11:
5:
1125:
1115:
1114:
1109:
1093:
1092:
1053:
1018:
999:(1): 157–171.
979:
934:(23): 231602.
918:
911:
885:
884:
882:
879:
876:
875:
871:chiral anomaly
862:
844:
843:
841:
838:
837:
836:
831:
826:
819:
816:
802:
799:
796:
793:
762:
742:
722:
702:
699:
696:
693:
673:
653:
650:
647:
644:
624:
604:
589:
586:
546:
543:
497:
493:
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480:
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471:
465:
461:
457:
454:
451:
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429:
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397:
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389:
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375:
371:
365:
361:
357:
354:
351:
348:
343:
339:
333:
329:
325:
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303:
292:
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283:
280:
265:
261:
257:
254:
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243:
239:
233:
229:
225:
222:
219:
216:
211:
207:
201:
197:
193:
190:
187:
176:Dirac fermions
161:
157:
136:
131:
127:
123:
120:
117:
95:
91:
48:
45:
33:chiral anomaly
15:
9:
6:
4:
3:
2:
1124:
1113:
1110:
1108:
1105:
1104:
1102:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1057:
1049:
1045:
1041:
1037:
1033:
1029:
1022:
1014:
1010:
1006:
1002:
998:
994:
990:
983:
975:
971:
967:
963:
959:
955:
951:
947:
942:
937:
933:
929:
922:
914:
908:
904:
900:
893:
891:
886:
872:
866:
856:
854:
852:
850:
845:
835:
832:
830:
829:'t Hooft loop
827:
825:
822:
821:
815:
797:
782:
780:
776:
760:
740:
720:
697:
671:
648:
622:
602:
593:
585:
582:
577:
575:
574:non-anomalous
571:
568:only to this
567:
563:
560:
556:
552:
542:
540:
536:
532:
528:
524:
520:
517:
513:
495:
487:
481:
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459:
452:
449:
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431:
427:
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417:
395:
387:
381:
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359:
352:
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341:
331:
327:
320:
317:
309:
302:
298:
288:
279:
263:
255:
249:
246:
241:
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227:
220:
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214:
209:
199:
195:
188:
185:
177:
159:
155:
129:
125:
118:
115:
93:
89:
80:
75:
73:
68:
66:
62:
60:
54:
44:
42:
38:
34:
30:
26:
22:
1070:
1066:
1056:
1031:
1027:
1021:
996:
992:
982:
931:
927:
921:
903:Plenum Press
898:
865:
783:
594:
591:
578:
548:
300:
294:
285:
76:
71:
69:
64:
58:
50:
40:
36:
24:
18:
570:gauge field
551:gauge field
57:chiral, or
1101:Categories
881:References
1073:(1): 95.
941:1403.0617
860:energies.
814:freedom.
792:Γ
692:Γ
643:Γ
479:×
447:×
379:×
347:×
306:massless
247:×
215:×
174:massless
61:anomalies
974:35171032
966:24972194
818:See also
562:fermions
527:fermions
519:fermions
516:massless
39:and the
1075:Bibcode
1036:Bibcode
1001:Bibcode
946:Bibcode
555:couples
291:Example
63:, and '
972:
964:
909:
566:couple
564:which
559:chiral
553:which
533:, the
308:quarks
23:, the
970:S2CID
936:arXiv
840:Notes
545:Proof
299:with
81:with
962:PMID
907:ISBN
1083:doi
1044:doi
1032:250
1009:doi
997:177
954:doi
932:112
531:QCD
521:or
27:by
19:In
1103::
1081:.
1071:37
1069:.
1065:.
1042:.
1030:.
1007:.
995:.
991:.
968:.
960:.
952:.
944:.
930:.
905:.
901:.
889:^
848:^
781:.
773:.
67:.
43:.
1089:.
1085::
1077::
1050:.
1046::
1038::
1015:.
1011::
1003::
976:.
956::
948::
938::
915:.
801:]
798:A
795:[
761:A
741:A
721:G
701:]
698:A
695:[
672:G
652:]
649:A
646:[
623:A
603:G
496:V
492:)
488:1
485:(
482:U
474:R
470:)
464:f
460:N
456:(
453:U
450:S
442:L
438:)
432:f
428:N
424:(
421:U
418:S
396:V
392:)
388:1
385:(
382:U
374:R
370:)
364:f
360:N
356:(
353:U
350:S
342:L
338:)
332:f
328:N
324:(
321:U
318:S
304:f
301:N
264:V
260:)
256:1
253:(
250:U
242:R
238:)
232:f
228:N
224:(
221:U
218:S
210:L
206:)
200:f
196:N
192:(
189:U
186:S
160:f
156:N
135:)
130:c
126:N
122:(
119:U
116:S
94:f
90:N
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