55:
466:
117:
This saddle point rests at the top of a barrier between two different low-energy equilibria of a given system; the two equilibria are labeled with two different baryon numbers. One of the equilibria might consist of three baryons; the other, alternative, equilibrium for the same system might consist
208:
In absence of processes which violate B − L it is possible for an initial baryon asymmetry to be protected if it has a non-zero projection onto B − L. In this case the sphaleron processes would impose an equilibrium which distributes the initial B asymmetry between
964:
from conversion of baryons to antileptons would be orders of magnitude higher than the energy efficiency of existing power-generation technology such as nuclear fusion. Tegmark speculates that an extremely advanced civilization might use a "sphalerizer" to generate energy from ordinary baryonic
311:
177:
Since a sphaleron may convert baryons to antileptons and antibaryons to leptons and thus change the baryon number, if the density of sphalerons was at some stage high enough, they could wipe out any net excess of baryons or anti-baryons. This has two important implications in any theory of
943:
and each of the lepton families is raised (or lowered, depending on the winding direction) by one; as there are three quark families, baryon number can only change in multiples of three. The baryon number violation can alternatively be visualized in terms of a kind of
126:-like process) or must for a reasonable period of time be brought up to a high enough energy that it can classically cross over the barrier (in which case the process is termed a "sphaleron" process and can be modeled with an eponymous sphaleron particle).
948:: in the course of the winding, a baryon originally considered to be part of the vacuum is now considered a real baryon, or vice versa, and all the other baryons stacked inside the sea are accordingly shifted by one energy level.
204:. This is because in a second-order phase transition, sphalerons would wipe out any baryon asymmetry as it is created, while in a first-order phase transition, sphalerons would wipe out baryon asymmetry only in the unbroken phase.
461:{\displaystyle \mathbf {A} =\nu \,{\frac {\,f(\xi )\,}{\xi }}~{\hat {\mathbf {r} }}\times \mathbf {\sigma } \,,\qquad \phi ={\frac {\nu }{\,{\sqrt {2\,}}\,}}~h(\xi )~{\hat {\mathbf {r} }}\cdot \mathbf {\sigma } ~\phi _{0}}
878:
129:
In both the instanton and sphaleron cases, the process can only convert groups of three baryons into three antileptons (or three antibaryons into three leptons) and vice versa. This violates conservation of
573:
1719:
1043:
Phong, Vo Quoc; Khiem, Phan Hong; Loc, Ngo Phuc Duc; Long, Hoang Ngoc (2020). "Sphaleron in the first-order electroweak phase transition with the dimension-six Higgs field operator".
833:
150:
collisions, because although the LHC can create collisions of energy 10 TeV and greater, the generated energy cannot be concentrated in a manner that would create sphalerons.
908:
511:
302:
251:
1363:
729:
694:
596:
775:
804:
755:
652:
939:
Baryon number violation is caused by the "winding" of the fields from one equilibrium to another. Each time the weak gauge fields wind, the count for each of the
934:
626:
165:. This means that under normal conditions sphalerons are unobservably rare. However, they would have been more common at the higher temperatures of the
1255:
Rubakov, Valery A.; Shaposhnikov, Mikhail E. (1996). "Electroweak baryon number nonconservation in the early universe and in high-energy collisions".
1096:
Papaefstathiou, Andreas; Plätzer, Simon; Sakurai, Kazuki (2019). "On the phenomenology of sphaleron-induced processes at the LHC and beyond".
1646:
Diakonov, Dmitri; Polyakov, Maxim; Sieber, Peter; Schaldach, Jörg; Goeke, Klaus (15 June 1994). "Fermion sea along the sphaleron barrier".
1595:
Arnold, Peter; McLerran, Larry (15 February 1988). "The sphaleron strikes back: A response to objections to the sphaleron approximation".
1452:
Kuzmin, V.A.; Rubakov, V.A.; Shaposhnikov, M.E. (1985). "On anomalous electroweak baryon-number non-conservation in the early universe".
838:
200:
While a baryon net excess can be created during the electroweak symmetry breaking, it can be preserved only if this phase transition was
142:
is conserved. The minimum energy required to trigger the sphaleron process is believed to be around 10 TeV; however, sphalerons
1729:
1311:
516:
209:
both B and L numbers. In some theories of baryogenesis, an imbalance of the number of leptons and antileptons is formed first by
1487:
Harvey, J.; Turner, M. (1990). "Cosmological baryon and lepton number in the presence of electroweak fermion-number violation".
940:
1755:
780:
For a sphaleron in the background of a non-broken phase, the Higgs field must obviously fall off eventually to zero as
1750:
1550:
Arnold, P.; McLerran, L. (1987). "Sphalerons, small fluctuations, and baryon-number violation in electroweak theory".
1023:
There is no true instanton in electroweak theory; instead, the tunneling rate is determined by constrained instantons.
961:
1149:
Zhou, Ruiyu; Bian, Ligong; Guo, Huai-Ke (2020). "Connecting the electroweak sphaleron with gravitational waves".
658:
812:
1391:
191:
79:
1389:
Shaposhnikov, M.E.; Farrar, G.R. (1993). "Baryon asymmetry of the universe in the minimal standard model".
887:
210:
197:
would be wiped out due to abundant sphalerons caused by high temperatures existing in the early universe.
147:
118:
of three antileptons. In order to cross this barrier and change the baryon number, a system must either
476:
213:
and sphaleron transitions then convert this to an imbalance in the numbers of baryons and antibaryons.
1665:
1604:
1561:
1498:
1461:
1410:
1336:
1221:
1168:
1115:
1062:
992:
261:
99:
1697:
1655:
1532:
1434:
1400:
1284:
1266:
1237:
1211:
1184:
1158:
1131:
1105:
1078:
1052:
229:
1327:
Klinkhamer, F.R.; Manton, N.S. (1984). "A saddle-point solution in the
Weinberg-Salam theory".
699:
664:
1725:
1689:
1681:
1628:
1620:
1577:
1552:
1524:
1489:
1426:
1307:
578:
119:
1202:
Ho, David L.-J.; Rajantie, Arttu (2020). "Electroweak sphaleron in a strong magnetic field".
760:
1673:
1612:
1569:
1514:
1506:
1469:
1418:
1344:
1276:
1229:
1176:
1123:
1070:
998:
783:
734:
631:
201:
162:
107:
87:
1257:
194:
103:
913:
605:
1669:
1608:
1565:
1502:
1465:
1414:
1340:
1225:
1172:
1119:
1066:
974:
881:
226:
183:
83:
71:
43:
1744:
1473:
1288:
1241:
1188:
1135:
1082:
135:
131:
1701:
1536:
1438:
1280:
986:
731:, which must be determined numerically, go from 0 to 1 in value as their argument,
179:
111:
59:
1715:
1422:
1233:
1180:
1127:
1074:
957:
655:
255:
1685:
1677:
1624:
1616:
1510:
1348:
936:, hence establishing the connection between the sphaleron and the instanton.
1573:
1519:
980:
945:
123:
1693:
1528:
1430:
1632:
1581:
54:
166:
25:
139:
1660:
1405:
1364:"Think of the universe as a skateboard park: Supernovas and sphalerons"
1271:
30:
Roughly, a high-energy composite of 3 leptons or of 3 baryons
95:
91:
114:
of the electroweak potential (in infinite-dimensional field space).
1216:
1163:
1110:
1057:
873:{\displaystyle {\frac {~\mathbf {r} \cdot \mathbf {\sigma } ~}{r}}}
835:, the gauge sector approaches one of the pure-gauge transformation
880:, which is the same as the pure gauge transformation to which the
599:
222:
53:
90:, and is involved in certain hypothetical processes that violate
1302:
White, Graham Albert (2016). "Section 3.5: The sphaleron".
568:{\displaystyle ~\phi _{0}={\begin{bmatrix}1\\0\end{bmatrix}}~}
254:, we have the following equations for the gauge field and the
16:
Solution to field equations in
Standard Model particle physics
1721:
Life 3.0: Being Human in the Age of
Artificial Intelligence
78:"slippery") is a static (time-independent) solution to the
122:
through the barrier (in which case the transition is an
1304:
995: – Finite energy solutions in Euclidean spacetime
541:
977: – Non-conservation of chiral current in physics
916:
890:
841:
815:
786:
763:
737:
702:
667:
634:
608:
581:
519:
479:
314:
264:
232:
42:
34:
24:
1654:(12). American Physical Society (APS): 6864–6882.
928:
902:
872:
827:
798:
769:
749:
723:
688:
646:
620:
590:
567:
505:
460:
296:
245:
1603:(4). American Physical Society (APS): 1020–1029.
98:numbers. Such processes cannot be represented by
8:
19:
1718:(2017). "Chapter 6: Our cosmic endowment".
628:is the electroweak coupling constant, and
1659:
1518:
1404:
1270:
1215:
1162:
1109:
1056:
915:
889:
856:
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315:
313:
282:
269:
263:
237:
231:
190:Any baryon net excess arising before the
1035:
1016:
983: – Solitons in Euclidean spacetime
828:{\displaystyle \xi \rightarrow \infty }
153:A sphaleron is similar to the midpoint
989: – Yang–Mills theory vacuum state
18:
7:
1362:Butterworth, Jon (8 November 2016).
1306:. Morgan & Claypool Publishers.
1001: – Solutions of Lamé's equation
903:{\displaystyle r\rightarrow \infty }
897:
822:
764:
110:. Geometrically, a sphaleron is a
14:
506:{\displaystyle ~\xi =r\,g\,\nu ~}
849:
427:
357:
316:
1281:10.1070/PU1996v039n05ABEH000145
379:
1098:Journal of High Energy Physics
894:
819:
715:
709:
680:
674:
661:absolute value. The functions
431:
417:
411:
361:
340:
334:
62:(in red) on a simple function.
1:
297:{\displaystyle A_{0}=A_{r}=0}
1474:10.1016/0370-2693(85)91028-7
598:represent the generators of
1423:10.1103/PhysRevLett.70.2833
1234:10.1103/PhysRevD.102.053002
1181:10.1103/PhysRevD.101.091903
1075:10.1103/PhysRevD.101.116010
246:{\displaystyle \theta _{W}}
161:of the instanton, so it is
106:, and are therefore called
80:electroweak field equations
1772:
724:{\displaystyle ~f(\xi )~}
689:{\displaystyle ~h(\xi )~}
225:gauge theory, neglecting
75:
1724:(Kindle 3839 ed.).
1678:10.1103/physrevd.49.6864
1617:10.1103/physrevd.37.1020
1511:10.1103/PhysRevD.42.3344
1349:10.1103/PhysRevD.30.2212
591:{\displaystyle ~\sigma }
146:be produced in existing
1574:10.1103/PhysRevD.36.581
1392:Physical Review Letters
1128:10.1007/JHEP12(2019)017
956:According to physicist
809:Note that in the limit
770:{\displaystyle \infty }
930:
904:
874:
829:
800:
799:{\displaystyle ~\xi ~}
771:
751:
750:{\displaystyle ~\xi ~}
725:
690:
648:
647:{\displaystyle ~\nu ~}
622:
592:
569:
507:
462:
298:
247:
63:
931:
905:
875:
830:
801:
772:
752:
726:
691:
649:
623:
593:
570:
508:
463:
299:
248:
140:B − L
138:, but the difference
57:
1006:References and notes
914:
888:
839:
813:
784:
761:
735:
700:
665:
632:
606:
579:
517:
477:
312:
262:
230:
100:perturbative methods
1756:Anomalies (physics)
1670:1994PhRvD..49.6864D
1609:1988PhRvD..37.1020A
1566:1987PhRvD..36..581A
1503:1990PhRvD..42.3344H
1466:1985PhLB..155...36K
1415:1993PhRvL..70.2833F
1341:1984PhRvD..30.2212K
1226:2020PhRvD.102e3002H
1173:2020PhRvD.101i1903Z
1120:2019JHEP...12..017P
1067:2020PhRvD.101k6010P
993:Periodic instantons
929:{\displaystyle t=0}
621:{\displaystyle ~g~}
21:
1751:Electroweak theory
960:, the theoretical
926:
900:
870:
825:
806:goes to infinity.
796:
767:
747:
721:
686:
644:
618:
588:
565:
556:
503:
458:
294:
243:
64:
1648:Physical Review D
1597:Physical Review D
1553:Physical Review D
1497:(10): 3344–3349.
1490:Physical Review D
1454:Physics Letters B
1399:(19): 2833–2836.
1335:(10): 2212–2220.
1329:Physical Review D
1204:Physical Review D
1151:Physical Review D
1045:Physical Review D
962:energy efficiency
868:
863:
847:
795:
789:
757:, goes from 0 to
746:
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720:
705:
685:
670:
643:
637:
617:
611:
584:
564:
522:
502:
482:
447:
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407:
403:
399:
364:
352:
348:
195:symmetry breaking
52:
51:
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1705:
1663:
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1637:
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1541:
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1522:
1520:2060/19900014807
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274:
273:
252:
250:
249:
244:
242:
241:
163:non-perturbative
160:
158:
108:non-perturbative
104:Feynman diagrams
88:particle physics
77:
58:An example of a
22:
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1770:
1766:
1765:
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1371:
1361:
1360:
1356:
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1314:
1301:
1300:
1296:
1258:Physics-Uspekhi
1254:
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1743:
1742:
1738:
1737:
1730:
1707:
1661:hep-ph/9311374
1638:
1587:
1560:(2): 581–596.
1542:
1479:
1460:(1–2): 36–42.
1444:
1406:hep-ph/9305274
1381:
1354:
1319:
1312:
1294:
1272:hep-ph/9603208
1265:(5): 461–502.
1247:
1194:
1141:
1088:
1051:(11): 116010.
1034:
1033:
1032:
1026:
1025:
1015:
1014:
1013:
1012:
1007:
1004:
1003:
1002:
996:
990:
984:
978:
975:Chiral anomaly
970:
967:
953:
952:Energy release
950:
941:quark families
925:
922:
919:
899:
896:
893:
884:approaches as
882:BPST instanton
867:
859:
855:
851:
824:
821:
818:
792:
766:
743:
717:
714:
711:
708:
682:
679:
676:
673:
640:
614:
587:
575:, the symbols
559:
553:
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184:Standard Model
174:
171:
167:early universe
84:Standard Model
50:
49:
46:
40:
39:
36:
32:
31:
28:
15:
13:
10:
9:
6:
4:
3:
2:
1768:
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1754:
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1731:9780451485090
1727:
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1313:9781681744582
1309:
1305:
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1278:
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1268:
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1260:
1259:
1251:
1248:
1243:
1239:
1235:
1231:
1227:
1223:
1218:
1213:
1210:(5): 053002.
1209:
1205:
1198:
1195:
1190:
1186:
1182:
1178:
1174:
1170:
1165:
1160:
1157:(9): 091903.
1156:
1152:
1145:
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1137:
1133:
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1125:
1121:
1117:
1112:
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1103:
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1010:
1009:
1005:
1000:
999:Lamé function
997:
994:
991:
988:
985:
982:
979:
976:
973:
972:
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987:Theta vacuum
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220:
211:leptogenesis
207:
180:baryogenesis
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152:
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112:saddle point
67:
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60:saddle point
48:~10 TeV
38:Hypothetical
1366:. Science.
958:Max Tegmark
256:Higgs field
202:first-order
192:electroweak
182:within the
26:Composition
1745:Categories
1374:1 December
1217:2005.03125
1164:1910.00234
1111:1910.04761
1104:(12): 17.
1058:2003.09625
1686:0556-2821
1625:0556-2821
1289:250852429
1242:218538382
1189:203610139
1136:204401729
1083:214612257
1031:Citations
981:Instanton
946:Dirac sea
898:∞
895:→
858:σ
854:⋅
823:∞
820:→
817:ξ
791:ξ
765:∞
742:ξ
713:ξ
678:ξ
639:ν
586:σ
525:ϕ
498:ν
484:ξ
450:ϕ
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438:⋅
432:^
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389:ν
381:ϕ
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368:×
362:^
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124:instanton
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20:Sphaleron
1702:18303496
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1537:28823418
1529:10012733
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969:See also
965:matter.
102:such as
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1666:Bibcode
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96:lepton
92:baryon
35:Status
1698:S2CID
1656:arXiv
1533:S2CID
1435:S2CID
1401:arXiv
1285:S2CID
1267:arXiv
1238:S2CID
1212:arXiv
1185:S2CID
1159:arXiv
1132:S2CID
1106:arXiv
1079:S2CID
1053:arXiv
1011:Notes
656:Higgs
600:SU(2)
223:SU(2)
159:= 0 )
72:Greek
1726:ISBN
1690:PMID
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1525:PMID
1427:PMID
1376:2017
1370:. UK
1308:ISBN
1102:2019
696:and
134:and
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44:Mass
1674:doi
1613:doi
1570:doi
1515:hdl
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1345:doi
1277:doi
1230:doi
1208:102
1177:doi
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