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Nambu–Jona-Lasinio model

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2217: 25: 1330: 668: 1095: 465: 1424:. However, while it is able to model chiral symmetry breaking and chiral condensates, it does not model confinement. Also, the axial symmetry is broken spontaneously in this model, leading to a massless Goldstone boson unlike QCD, where it is broken anomalously. 1085: 452: 1325:{\displaystyle =\,i\,{\bar {\psi }}_{La}\partial \!\!\!/\psi _{L}^{a}+\,i\,{\bar {\psi }}_{Ra}\partial \!\!\!/\psi _{R}^{a}+{\frac {\lambda }{N}}\,\left({\bar {\psi }}_{La}\psi _{R}^{b}\right)\left({\bar {\psi }}_{Rb}\psi _{L}^{a}\right).} 663:{\displaystyle {\mathcal {L}}=\,i\,{\bar {\psi }}_{L}\partial \!\!\!/\psi _{L}+\,i\,{\bar {\psi }}_{R}\partial \!\!\!/\psi _{R}+\lambda \,\left({\bar {\psi }}_{L}\psi _{R}\right)\left({\bar {\psi }}_{R}\psi _{L}\right).} 1377:
acting upon the left-handed flavors and right-handed flavors respectively is the chiral symmetry (in other words, there is no natural correspondence between the left-handed and the right-handed flavors),
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and is defined in a spacetime with an even number of dimensions. It is still important and is used primarily as an effective although not rigorous low energy substitute for quantum chromodynamics.
810: 768: 252: 1647: 727: 694: 42: 2070: 1880: 1962: 1921: 1700: 165:. The "complicatedness" of the theory has become more natural as it is now seen as a low-energy approximation of the still more basic theory of 1725: 89: 1829: 2060: 61: 1952: 1640: 1394:) since the condensate leads to a pairing of the left-handed and the right-handed flavors. The axial charge is also spontaneously broken. 1808: 1690: 68: 2220: 1900: 1566: 108: 75: 2150: 1957: 1720: 1413: 1753: 2239: 2110: 2013: 1993: 1633: 1916: 1675: 1670: 198: 57: 46: 770:
which represents short-distance physics or the strong interaction scale, producing an attractive four-fermion interaction.
185:, also contributed essentially to the theory of superconductivity, i.e., by the "Nambu formalism". The second inventor was 782: 2189: 1715: 1550: 212:
The model is quite technical, although based essentially on symmetry principles. It is an example of the importance of
2184: 1978: 1758: 1584:"On the application of the methods of superconductivity theory to the problem of the masses of elementary particles" 2244: 1926: 35: 213: 2162: 190: 2034: 1859: 1834: 1337:
Chiral symmetry forbids a bare mass term, but there may be chiral condensates. The global symmetry here is SU(
82: 1390:. If a chiral condensate forms, then the chiral symmetry is spontaneously broken into a diagonal subgroup SU( 1080:{\displaystyle {\mathcal {L}}=\,i\,{\bar {\psi }}_{a}\partial \!\!\!/\psi ^{a}+{\frac {\lambda }{4N}}\,\left} 1890: 1773: 697:
are an attractive four-fermion interaction, which parallels the BCS theory phonon exchange interaction. The
2201: 2125: 2120: 2055: 1983: 1885: 1788: 1778: 1763: 1680: 1432: 1417: 1398: 189:. The common paper of the authors that introduced the model appeared in 1961. A subsequent paper included 166: 1936: 1839: 1803: 1748: 1448: 2174: 2169: 2008: 2003: 1793: 1611: 228: 186: 732: 181:, particularly from the BCS breakthrough of 1957. The first inventor of the Nambu–Jona-Lasinio model, 2115: 1849: 1685: 1656: 1522: 1481: 232: 224: 220: 122: 447:{\displaystyle {\mathcal {L}}=\,i\,{\bar {\psi }}\partial \!\!\!/\psi +{\frac {\lambda }{4}}\,\left} 2179: 2145: 2080: 1998: 1988: 1783: 202: 1813: 243: 178: 1619: 1864: 1562: 1554: 1428: 778: 774: 712: 679: 162: 2135: 1530: 1489: 773:
There is no bare fermion mass term because of the chiral symmetry. However, there will be a
239: 2157: 1895: 1844: 1421: 1406: 698: 146: 1526: 1511:"Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II" 1485: 2100: 2075: 1854: 1615: 1470:"Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I" 206: 182: 1583: 2233: 2196: 2130: 2105: 1705: 1387: 201:. At the same time, the same model was independently considered by Soviet physicists 142: 2140: 2050: 1768: 1710: 1695: 1798: 1436: 709:
where Q is the ordinary charge of the Dirac fermion and χ is the chiral charge.
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The Rise of the Standard Model: A History of Particle Physics from 1964 to 1979
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is the Dirac charge, which is sometimes called the baryon number and U(1)
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from fermion interactions inspired many theories of the breaking of
1622:, Scholarpedia, 5(12):7487, (2010). doi:10.4249/scholarpedia.7487 138: 1402: 792:
flavors and the flavor indices represented by the Latin letters
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in four spacetime dimensions, this theory can only be an
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The model is much inspired by the different field of
169:, which does not work perturbatively at low energies. 1098: 813: 785:
of the chiral symmetry, but not the charge symmetry.
735: 715: 682: 468: 255: 2093: 2043: 2022: 1971: 1945: 1909: 1873: 1822: 1741: 1734: 1663: 49:. Unsourced material may be challenged and removed. 1324: 1079: 762: 721: 688: 662: 446: 1190: 1189: 1188: 1134: 1133: 1132: 853: 852: 851: 556: 555: 554: 508: 507: 506: 288: 287: 286: 1412:As mentioned, this model is sometimes used as a 1641: 8: 1509:Nambu, Y.; Jona-Lasinio, G. (October 1961). 1561:. Cambridge University Press. p. 244. 1738: 1648: 1634: 1626: 1468:Nambu, Y.; Jona-Lasinio, G. (April 1961). 781:) leading to an effective mass term and a 1534: 1493: 1308: 1303: 1290: 1279: 1278: 1261: 1256: 1243: 1232: 1231: 1224: 1214: 1205: 1200: 1191: 1176: 1165: 1164: 1162: 1158: 1149: 1144: 1135: 1120: 1109: 1108: 1106: 1102: 1097: 1061: 1051: 1041: 1030: 1029: 1012: 1002: 992: 981: 980: 960: 950: 939: 938: 921: 911: 900: 899: 887: 872: 863: 854: 842: 831: 830: 828: 824: 815: 814: 812: 754: 745: 734: 714: 681: 646: 636: 625: 624: 607: 597: 586: 585: 578: 566: 557: 545: 534: 533: 531: 527: 518: 509: 497: 486: 485: 483: 479: 470: 469: 467: 425: 410: 409: 390: 375: 374: 347: 346: 322: 321: 310: 300: 289: 272: 271: 270: 266: 257: 256: 254: 109:Learn how and when to remove this message 1460: 1397:The broken symmetries lead to massless 133:) is a complicated effective theory of 1726:Two-dimensional conformal field theory 1427:Since the Nambu–Jona-Lasinio model is 729:is actually an inverse squared mass, 7: 47:adding citations to reliable sources 1582:Vaks, V. G.; Larkin, A. I. (1961). 1401:bosons which are sometimes called 1185: 1129: 848: 551: 503: 283: 149:, paralleling the construction of 16:Effective field theory of nucleons 14: 2221:Template:Quantum mechanics topics 804:, the Lagrangian density becomes 2216: 2215: 763:{\displaystyle \lambda =1/M^{2}} 131:the Nambu and Jona-Lasinio model 23: 34:needs additional citations for 1284: 1237: 1170: 1114: 1035: 986: 944: 905: 836: 630: 591: 539: 491: 415: 380: 352: 327: 277: 1: 783:spontaneous symmetry breaking 141:constructed from interacting 1555:"13. A View from the Island" 219:The dynamical creation of a 2185:Quantum information science 2261: 675:The terms proportional to 58:"Nambu–Jona-Lasinio model" 2210: 214:four-fermion interactions 1620:Nambu-Jona-Lasinio model 722:{\displaystyle \lambda } 689:{\displaystyle \lambda } 191:chiral symmetry breaking 127:Nambu–Jona-Lasinio model 1881:2D free massless scalar 1774:Quantum electrodynamics 1701:QFT in curved spacetime 1536:10.1103/PhysRev.124.246 1495:10.1103/PhysRev.122.345 2240:Quantum chromodynamics 2202:Quantum thermodynamics 2126:On shell and off shell 2121:Loop quantum cosmology 1963:N = 4 super Yang–Mills 1922:N = 1 super Yang–Mills 1789:Scalar electrodynamics 1779:Quantum chromodynamics 1681:Conformal field theory 1657:Quantum field theories 1433:effective field theory 1418:quantum chromodynamics 1414:phenomenological model 1326: 1081: 764: 723: 690: 664: 448: 238:Starting with the one- 167:quantum chromodynamics 2175:Quantum hydrodynamics 2170:Quantum hadrodynamics 1794:Scalar chromodynamics 1612:Giovanni Jona-Lasinio 1327: 1082: 765: 724: 691: 665: 449: 187:Giovanni Jona-Lasinio 2146:Quantum fluctuations 2116:Loop quantum gravity 1686:Lattice field theory 1096: 811: 733: 713: 701:of the model is U(1) 680: 466: 253: 233:top-quark condensate 225:electroweak symmetry 129:(or more precisely: 123:quantum field theory 43:improve this article 2180:Quantum information 1784:Quartic interaction 1527:1961PhRv..124..246N 1486:1961PhRv..122..345N 1313: 1266: 1210: 1154: 2066:Nambu–Jona-Lasinio 1994:Higher dimensional 1901:Wess–Zumino–Witten 1691:Noncommutative QFT 1551:Alexander Polyakov 1435:which needs to be 1322: 1299: 1252: 1196: 1140: 1077: 760: 719: 686: 660: 457:or, equivalently, 444: 244:Lagrangian density 179:solid state theory 2245:Superconductivity 2227: 2226: 2089: 2088: 1449:Gross–Neveu model 1429:nonrenormalizable 1287: 1240: 1222: 1173: 1117: 1038: 989: 947: 908: 885: 839: 775:chiral condensate 633: 594: 542: 494: 418: 383: 355: 330: 308: 280: 163:superconductivity 119: 118: 111: 93: 2252: 2219: 2218: 2136:Quantum dynamics 1809:Yang–Mills–Higgs 1764:Non-linear sigma 1754:Euler–Heisenberg 1739: 1650: 1643: 1636: 1627: 1599: 1598: 1588: 1579: 1573: 1572: 1547: 1541: 1540: 1538: 1506: 1500: 1499: 1497: 1465: 1331: 1329: 1328: 1323: 1318: 1314: 1312: 1307: 1298: 1297: 1289: 1288: 1280: 1271: 1267: 1265: 1260: 1251: 1250: 1242: 1241: 1233: 1223: 1215: 1209: 1204: 1195: 1184: 1183: 1175: 1174: 1166: 1153: 1148: 1139: 1128: 1127: 1119: 1118: 1110: 1086: 1084: 1083: 1078: 1076: 1072: 1071: 1067: 1066: 1065: 1056: 1055: 1046: 1045: 1040: 1039: 1031: 1022: 1018: 1017: 1016: 1007: 1006: 997: 996: 991: 990: 982: 970: 966: 965: 964: 955: 954: 949: 948: 940: 931: 927: 926: 925: 916: 915: 910: 909: 901: 886: 884: 873: 868: 867: 858: 847: 846: 841: 840: 832: 820: 819: 769: 767: 766: 761: 759: 758: 749: 728: 726: 725: 720: 695: 693: 692: 687: 669: 667: 666: 661: 656: 652: 651: 650: 641: 640: 635: 634: 626: 617: 613: 612: 611: 602: 601: 596: 595: 587: 571: 570: 561: 550: 549: 544: 543: 535: 523: 522: 513: 502: 501: 496: 495: 487: 475: 474: 453: 451: 450: 445: 443: 439: 438: 434: 430: 429: 420: 419: 411: 403: 399: 395: 394: 385: 384: 376: 365: 361: 357: 356: 348: 340: 336: 332: 331: 323: 309: 301: 293: 282: 281: 273: 262: 261: 242:case first, the 114: 107: 103: 100: 94: 92: 51: 27: 19: 2260: 2259: 2255: 2254: 2253: 2251: 2250: 2249: 2230: 2229: 2228: 2223: 2206: 2158:Quantum gravity 2085: 2044:Particle theory 2039: 2018: 1967: 1941: 1905: 1869: 1823:Low dimensional 1818: 1759:Ginzburg–Landau 1730: 1721:Topological QFT 1659: 1654: 1608: 1603: 1602: 1591:Sov. Phys. JETP 1586: 1581: 1580: 1576: 1569: 1549: 1548: 1544: 1515:Physical Review 1508: 1507: 1503: 1474:Physical Review 1467: 1466: 1462: 1457: 1445: 1407:Goldstone boson 1385: 1381: 1376: 1368: 1360: 1356: 1352: 1344: 1277: 1276: 1272: 1230: 1229: 1225: 1163: 1107: 1094: 1093: 1057: 1047: 1028: 1027: 1023: 1008: 998: 979: 978: 974: 956: 937: 936: 932: 917: 898: 897: 893: 892: 888: 877: 859: 829: 809: 808: 750: 731: 730: 711: 710: 708: 704: 699:global symmetry 678: 677: 642: 623: 622: 618: 603: 584: 583: 579: 562: 532: 514: 484: 464: 463: 421: 408: 404: 386: 373: 369: 345: 341: 320: 316: 315: 311: 251: 250: 175: 147:chiral symmetry 115: 104: 98: 95: 52: 50: 40: 28: 17: 12: 11: 5: 2258: 2256: 2248: 2247: 2242: 2232: 2231: 2225: 2224: 2211: 2208: 2207: 2205: 2204: 2199: 2194: 2193: 2192: 2182: 2177: 2172: 2167: 2166: 2165: 2155: 2154: 2153: 2143: 2138: 2133: 2128: 2123: 2118: 2113: 2108: 2103: 2101:Casimir effect 2097: 2095: 2091: 2090: 2087: 2086: 2084: 2083: 2078: 2076:Standard Model 2073: 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183:Yoichiro Nambu 174: 171: 143:Dirac fermions 117: 116: 31: 29: 22: 15: 13: 10: 9: 6: 4: 3: 2: 2257: 2246: 2243: 2241: 2238: 2237: 2235: 2222: 2214: 2209: 2203: 2200: 2198: 2197:Quantum logic 2195: 2191: 2188: 2187: 2186: 2183: 2181: 2178: 2176: 2173: 2171: 2168: 2164: 2161: 2160: 2159: 2156: 2152: 2149: 2148: 2147: 2144: 2142: 2139: 2137: 2134: 2132: 2131:Quantum chaos 2129: 2127: 2124: 2122: 2119: 2117: 2114: 2112: 2109: 2107: 2106:Cosmic string 2104: 2102: 2099: 2098: 2096: 2092: 2082: 2079: 2077: 2074: 2072: 2069: 2067: 2064: 2062: 2059: 2057: 2054: 2052: 2049: 2048: 2046: 2042: 2036: 2033: 2031: 2028: 2027: 2025: 2021: 2015: 2012: 2010: 2007: 2005: 2002: 2000: 1997: 1995: 1992: 1990: 1987: 1985: 1982: 1980: 1979:Pure 4D N = 1 1977: 1976: 1974: 1970: 1964: 1961: 1959: 1956: 1954: 1951: 1950: 1948: 1944: 1938: 1935: 1933: 1930: 1928: 1925: 1923: 1920: 1918: 1915: 1914: 1912: 1908: 1902: 1899: 1897: 1894: 1892: 1889: 1887: 1884: 1882: 1879: 1878: 1876: 1872: 1866: 1863: 1861: 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203:Valentin Vaks 200: 196: 192: 188: 184: 180: 172: 170: 168: 164: 160: 156: 152: 148: 144: 140: 136: 132: 128: 124: 113: 110: 102: 91: 88: 84: 81: 77: 74: 70: 67: 63: 60: –  59: 55: 54:Find sources: 48: 44: 38: 37: 32:This article 30: 26: 21: 20: 2212: 2141:Quantum foam 2081:Stueckelberg 2065: 2035:Chern–Simons 1972:Supergravity 1711:Supergravity 1696:Gauge theory 1594: 1590: 1577: 1558: 1545: 1518: 1514: 1504: 1477: 1473: 1463: 1437:UV completed 1426: 1422:chiral limit 1411: 1399:pseudoscalar 1396: 1391: 1388:axial charge 1370: 1362: 1346: 1338: 1336: 801: 797: 793: 789: 787: 772: 676: 674: 456: 237: 218: 211: 176: 151:Cooper pairs 130: 126: 120: 105: 96: 86: 79: 72: 65: 53: 41:Please help 36:verification 33: 2023:Topological 1937:Wess–Zumino 1850:Sine-Gordon 1840:Gross–Neveu 1749:Born–Infeld 1716:Thermal QFT 1357:× U(1) 1353:× U(1) 779:confinement 229:technicolor 199:strangeness 99:August 2009 2234:Categories 1804:Yang–Mills 1597:: 192–193. 1455:References 705:×U(1) 227:, such as 221:condensate 159:BCS theory 69:newspapers 2213:See also: 1932:Super QCD 1886:Liouville 1874:Conformal 1845:Schwinger 1369:×SU( 1361:where SU( 1345:×SU( 1301:ψ 1285:¯ 1282:ψ 1254:ψ 1238:¯ 1235:ψ 1217:λ 1198:ψ 1186:∂ 1171:¯ 1168:ψ 1142:ψ 1130:∂ 1115:¯ 1112:ψ 1059:ψ 1049:γ 1036:¯ 1033:ψ 1010:ψ 1000:γ 987:¯ 984:ψ 972:− 958:ψ 945:¯ 942:ψ 919:ψ 906:¯ 903:ψ 875:λ 861:ψ 849:∂ 837:¯ 834:ψ 737:λ 717:λ 684:λ 644:ψ 631:¯ 628:ψ 605:ψ 592:¯ 589:ψ 576:λ 564:ψ 552:∂ 540:¯ 537:ψ 516:ψ 504:∂ 492:¯ 489:ψ 432:ψ 423:γ 416:¯ 413:ψ 397:ψ 388:γ 381:¯ 378:ψ 367:− 359:ψ 353:¯ 350:ψ 334:ψ 328:¯ 325:ψ 303:λ 295:ψ 284:∂ 278:¯ 275:ψ 155:electrons 2009:Type IIB 2004:Type IIA 1989:4D N = 8 1984:4D N = 1 1953:6D (2,0) 1917:4D N = 1 1896:Polyakov 1855:Thirring 1664:Theories 1553:(1997). 1443:See also 777:(but no 231:and the 173:Overview 135:nucleons 2111:History 2094:Related 1891:Minimal 1742:Regular 1523:Bibcode 1482:Bibcode 1420:in the 1386:is the 195:isospin 157:in the 83:scholar 2051:Chiral 1999:Type I 1814:Yukawa 1735:Models 1565:  1405:. See 240:flavor 139:mesons 125:, the 85:  78:  71:  64:  56:  2190:links 2163:links 2151:links 2071:NMSSM 2056:Fermi 1799:Soler 1769:Proca 1587:(PDF) 1403:pions 788:With 153:from 145:with 90:JSTOR 76:books 2061:MSSM 1958:ABJM 1865:Toda 1614:and 1563:ISBN 1378:U(1) 205:and 197:and 137:and 62:news 2014:11D 1531:doi 1519:124 1490:doi 1478:122 1416:of 246:is 161:of 121:In 45:by 2236:: 2030:BF 1618:, 1595:13 1593:. 1589:. 1557:. 1529:. 1517:. 1513:. 1488:. 1476:. 1472:. 1439:. 1409:. 800:, 796:, 235:. 209:. 193:, 1649:e 1642:t 1635:v 1571:. 1539:. 1533:: 1525:: 1498:. 1492:: 1484:: 1392:N 1384:χ 1380:Q 1375:R 1373:) 1371:N 1367:L 1365:) 1363:N 1359:χ 1355:Q 1351:R 1349:) 1347:N 1343:L 1341:) 1339:N 1320:. 1316:) 1310:a 1305:L 1295:b 1292:R 1274:( 1269:) 1263:b 1258:R 1248:a 1245:L 1227:( 1220:N 1212:+ 1207:a 1202:R 1193:/ 1181:a 1178:R 1160:i 1156:+ 1151:a 1146:L 1137:/ 1125:a 1122:L 1104:i 1100:= 1074:] 1069:) 1063:a 1053:5 1043:b 1025:( 1020:) 1014:b 1004:5 994:a 976:( 968:) 962:a 952:b 934:( 929:) 923:b 913:a 895:( 890:[ 882:N 879:4 870:+ 865:a 856:/ 844:a 826:i 822:= 817:L 802:c 798:b 794:a 790:N 756:2 752:M 747:/ 743:1 740:= 707:χ 703:Q 658:. 654:) 648:L 638:R 620:( 615:) 609:R 599:L 581:( 573:+ 568:R 559:/ 547:R 529:i 525:+ 520:L 511:/ 499:L 481:i 477:= 472:L 441:] 436:) 427:5 406:( 401:) 392:5 371:( 363:) 343:( 338:) 318:( 313:[ 306:4 298:+ 291:/ 268:i 264:= 259:L 112:) 106:( 101:) 97:( 87:· 80:· 73:· 66:· 39:.

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"Nambu–Jona-Lasinio model"
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quantum field theory
nucleons
mesons
Dirac fermions
chiral symmetry
Cooper pairs
electrons
BCS theory
superconductivity
quantum chromodynamics
solid state theory
Yoichiro Nambu
Giovanni Jona-Lasinio
chiral symmetry breaking
isospin
strangeness
Valentin Vaks
Anatoly Larkin
four-fermion interactions

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