1167:, where this quantum theory may be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the field theory has no interactions. The Landau pole question is generally considered to be of minor academic interest for quantum electrodynamics because of the inaccessibly large momentum scale at which the inconsistency appears. This is not however the case in theories that involve the elementary scalar Higgs boson, as the momentum scale at which a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at the
31:
1212:
1138:
can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when
1194:
The situation becomes more complex in theories that involve other particles however. In fact, the addition of other particles can turn a trivial theory into a nontrivial one, at the cost of introducing constraints. Depending on the details of the theory, the Higgs mass can be bounded or even
1269:
This approach covered the conceptual point and was given full computational substance in Wilson's extensive important contributions. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the
2833:
is nontrivial remains a serious unresolved question. Theoretical proofs of triviality of the pure scalar field theory exist, but the situation for the full standard model is unknown. The implied constraints on the standard model have been discussed.
2141:
1938:
1195:
calculable. These quantum triviality constraints are in sharp contrast to the picture one derives at the classical level, where the Higgs mass is a free parameter. Quantum triviality can also lead to a calculable Higgs mass in
1147:
is trivial in four spacetime dimensions, but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the
1767:
1625:
1441:
1635:. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be
2596:
2476:
1191:. If realistic models of particle physics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle may have to be modified or abandoned.
1651:
The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, and
Khalatnikov by finding the following relation of the observable charge
1196:
2760:
2376:
1564:
1477:
2402:
2502:
2622:
2247:
2434:
2324:
1365:
1332:
2805:
2698:
2727:
2671:
2551:
2279:
2971:
1504:
2030:
2642:
2522:
1848:
1524:
1299:
1254:. A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional
3031:
L. D. Landau; A. A. Abrikosov & I. M. Khalatnikov (1954). "Asymptotic
Expressin for the Green's Function of the Electron in Quantum Electrodynamics".
3052:
L. D. Landau; A. A. Abrikosov & I. M. Khalatnikov (1954). "Asymptotic
Expressin for the Green's Function of the Photon in Quantum Electrodynamics".
1677:
3153:
1117:
200:
1368:
240:
3341:
Urs Heller, Markus
Klomfass, Herbert Neuberger, and Pavlos Vranas, (1993). "Numerical analysis of the Higgs mass triviality bound",
1139:
realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as
1274:, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and
2767:
The latter case corresponds to the quantum triviality in the full theory (beyond its perturbation context), as can be seen by
3086:
975:
1175:. In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the
1569:
1385:
1110:
2883:
1266:
is a way to define the components of the theory at large distances as aggregates of components at shorter distances.
1135:
745:
2559:
2439:
2292:. According to the classification by Bogoliubov and Shirkov, there are three qualitatively different situations:
1376:
43:
3364:
1640:
3374:
1103:
308:
157:
117:
3007:
845:
775:
147:
3379:
1223:
303:
245:
3299:
3369:
3054:
3033:
3012:
152:
2732:
2339:
1533:
1446:
955:
2381:
293:
3207:
2481:
780:
410:
255:
220:
2601:
770:
1164:
1055:
525:
333:
328:
190:
2225:
2700:
and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of
2410:
2300:
1643:, but the nature of the quantum field theories associated with these remains an open question.
1337:
1304:
925:
905:
405:
338:
250:
215:
2783:
2676:
2989:(1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys.
2703:
2647:
2527:
2255:
2020:
1281:
In more technical terms, let us assume that we have a theory described by a certain function
1243:
1168:
815:
675:
371:
235:
97:
3308:
3271:
3260:"Non-triviality of gauge theories with elementary scalars and upper bounds on Higgs masses"
3234:
3172:
3117:
3078:
3010:; I. M. Khalatnikov (1954). "On the Elimination of Infinities in Quantum Electrodynamics".
2949:
2918:
2875:
2768:
2136:{\displaystyle {\frac {dg}{d\ln \mu }}=\beta (g)=\beta _{2}g^{2}+\beta _{3}g^{3}+\ldots ~,}
1482:
1251:
1131:
1060:
840:
376:
230:
22:
1933:{\displaystyle g_{0}={\frac {g_{\text{obs}}}{1-\beta _{2}g_{\text{obs}}\ln \Lambda /m}}~.}
142:
8:
1070:
400:
3312:
3275:
3238:
3176:
3121:
2922:
785:
515:
3324:
3224:
3188:
3162:
2904:
2780:
is finite, the theory is internally inconsistent. The only way to avoid it, is to tend
2627:
2507:
1509:
1284:
1275:
1080:
1005:
935:
820:
805:
735:
700:
690:
650:
565:
455:
415:
205:
167:
122:
110:
87:
82:
1242:
Modern considerations of triviality are usually formulated in terms of the real-space
3328:
3283:
3192:
3129:
3108:
3082:
2986:
2930:
2879:
2843:
1247:
1035:
965:
850:
695:
630:
600:
570:
430:
425:
270:
195:
185:
77:
755:
3316:
3279:
3180:
3125:
2926:
2830:
1372:
1153:
1075:
990:
970:
940:
890:
715:
710:
705:
590:
575:
505:
210:
137:
127:
62:
1278:
in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.
2909:
1762:{\displaystyle g_{\text{obs}}={\frac {g_{0}}{1+\beta _{2}g_{0}\ln \Lambda /m}}~,}
1259:
1250:
and others. Investigations of triviality are usually performed in the context of
1020:
1015:
985:
980:
950:
910:
880:
860:
765:
660:
655:
620:
605:
580:
560:
540:
470:
460:
381:
343:
260:
225:
57:
35:
30:
1143:. Strong evidence supports the idea that a field theory involving only a scalar
2826:
1176:
1149:
1045:
995:
915:
900:
885:
865:
835:
830:
825:
810:
795:
760:
750:
680:
615:
585:
520:
485:
475:
450:
321:
288:
162:
3184:
1818:
In fact, the proper interpretation of Eq.1 consists in its inversion, so that
800:
3358:
2863:
1050:
1040:
1030:
1025:
945:
930:
730:
725:
555:
550:
510:
480:
445:
435:
366:
361:
3051:
3030:
3002:
3216:
2943:
1379:, etc. It must contain the whole description of the physics of the system.
1271:
1085:
1010:
895:
870:
855:
790:
670:
665:
545:
535:
530:
495:
490:
465:
420:
298:
265:
92:
72:
2001:) and the existence of the "Landau pole" in Eq.2 has no physical meaning.
3297:
Lindner, M. (1986). "Implications of triviality for the standard model".
1382:
Now we consider a certain blocking transformation of the state variables
1160:
1144:
1065:
1000:
960:
920:
875:
740:
645:
640:
625:
610:
595:
2872:
Random Walks, Critical
Phenomena, and Triviality in Quantum Field Theory
1211:
3320:
3003:
2867:
720:
635:
500:
440:
132:
67:
2907:(1988). "Triviality Pursuit: Can Elementary Scalar Particles Exist?".
1262:'s paper in 1966 proposed the "block-spin" renormalization group. The
1156:, the question of triviality in Higgs models is of great importance.
3259:
3103:
2861:
1184:
685:
3229:
3167:
3220:
1566:. If this is achievable by a certain change in the parameters,
1180:
3072:
1202:
1188:
1172:
3209:
Mapping theorem and Green functions in Yang-Mills theory
1631:. The most important information in the RG flow are its
2952:
2786:
2735:
2706:
2679:
2650:
2630:
2604:
2562:
2530:
2510:
2484:
2442:
2413:
2384:
2342:
2303:
2258:
2228:
2033:
1851:
1680:
1572:
1536:
1512:
1485:
1449:
1388:
1340:
1307:
1287:
2965:
2799:
2754:
2721:
2692:
2665:
2636:
2616:
2590:
2545:
2516:
2496:
2470:
2428:
2396:
2370:
2318:
2273:
2241:
2135:
1932:
1761:
1619:
1558:
1518:
1498:
1471:
1435:
1359:
1326:
1293:
3143:I. M. Suslov (2010). "Asymptotic Behavior of the
3101:
2903:
1620:{\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}
1436:{\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}
3356:
3154:Journal of Experimental and Theoretical Physics
3104:"Is the standard model Higgs mass predictable?"
1811:tends to zero in the limit of infinite cut-off
1639:. Numerous fixed points appear in the study of
3151:Theory: A Scheme Without Complex Parameters".
3075:Introduction to the Theory of Quantized Fields
1258:theories, came from condensed matter physics.
16:Possible outcome of renormalization in physics
1111:
3142:
2946:(1966): "Scaling laws for Ising models near
2591:{\displaystyle \beta (g)\propto g^{\alpha }}
2471:{\displaystyle \beta (g)\propto g^{\alpha }}
1614:
1592:
1586:
1573:
1430:
1408:
1402:
1389:
1354:
1341:
1321:
1308:
2899:
2897:
2895:
1118:
1104:
29:
3228:
3166:
2825:As a result, the question of whether the
2281:relies on the appearance of the function
3257:
3073:N. N. Bogoliubov; D. V. Shirkov (1980).
2807:to infinity, which is possible only for
1646:
1334:and a certain set of coupling constants
1203:Triviality and the renormalization group
1159:This Higgs triviality is similar to the
3296:
3223:: Proceedings of Science. p. 039.
2892:
2171:) if it is integrated under conditions
1831:) is chosen to give a correct value of
3357:
3205:
3102:Callaway, D.; Petronzio, R. (1987).
2973:", Physics (Long Island City, N.Y.)
2249:is retained in the right hand side.
2024:
2015:as a function of the momentum scale
1842:
1671:
1206:
13:
2436:is non-alternating and behaves as
2391:
2004:The actual behavior of the charge
1910:
1739:
14:
3391:
1789:is the mass of the particle, and
1479:must be lower than the number of
2755:{\displaystyle \mu >\mu _{0}}
2371:{\displaystyle g(\mu )\to g^{*}}
1627:, then the theory is said to be
1559:{\displaystyle {\tilde {s}}_{i}}
1506:. Now let us try to rewrite the
1472:{\displaystyle {\tilde {s}}_{i}}
1210:
3335:
3290:
3251:
3199:
3136:
2397:{\displaystyle \mu \to \infty }
2326:has a zero at the finite value
3095:
3066:
3045:
3024:
2996:
2980:
2937:
2855:
2820:
2716:
2710:
2660:
2654:
2572:
2566:
2540:
2534:
2452:
2446:
2423:
2417:
2388:
2355:
2352:
2346:
2313:
2307:
2268:
2262:
2072:
2066:
1991:(since they were obtained for
1602:
1589:
1544:
1457:
1418:
1405:
1:
2849:
2673:is divergent at finite value
2497:{\displaystyle \alpha \leq 1}
1827:(related to the length scale
158:Spontaneous symmetry breaking
118:Symmetry in quantum mechanics
3284:10.1016/0550-3213(84)90410-3
3130:10.1016/0550-3213(87)90657-2
2931:10.1016/0370-1573(88)90008-7
2617:{\displaystyle \alpha >1}
1793:is the momentum cut-off. If
7:
3258:Callaway, D. J. E. (1984).
2837:
2167:
2161:
2149:
1977:
1971:
1946:
1775:
10:
3396:
3055:Doklady Akademii Nauk SSSR
3034:Doklady Akademii Nauk SSSR
3013:Doklady Akademii Nauk SSSR
2242:{\displaystyle \beta _{2}}
2222:, when only the term with
2019:is determined by the full
153:Explicit symmetry breaking
3185:10.1134/S1063776110090153
2429:{\displaystyle \beta (g)}
2319:{\displaystyle \beta (g)}
1367:. This function may be a
1360:{\displaystyle \{J_{k}\}}
1327:{\displaystyle \{s_{i}\}}
1183:masses like those of the
309:BargmannâWigner equations
3300:Zeitschrift fĂŒr Physik C
2800:{\displaystyle \mu _{0}}
2693:{\displaystyle \mu _{0}}
2252:The general behavior of
2722:{\displaystyle g(\mu )}
2666:{\displaystyle g(\mu )}
2546:{\displaystyle g(\mu )}
2274:{\displaystyle g(\mu )}
1660:with the "bare" charge
1301:of the state variables
1246:, largely developed by
1165:quantum electrodynamics
334:Electroweak interaction
329:Quantum electrodynamics
304:WheelerâDeWitt equation
191:Background field method
3206:Frasca, Marco (2011).
2967:
2801:
2756:
2723:
2694:
2667:
2638:
2618:
2592:
2553:continues to infinity;
2547:
2518:
2498:
2472:
2430:
2398:
2372:
2320:
2275:
2243:
2137:
2021:Gell-MannâLow equation
1934:
1763:
1641:lattice Higgs theories
1621:
1560:
1520:
1500:
1473:
1437:
1361:
1328:
1295:
339:Quantum chromodynamics
216:Effective field theory
3365:Renormalization group
3217:The many faces of QCD
3079:John Wiley & Sons
2968:
2966:{\displaystyle T_{c}}
2802:
2757:
2724:
2695:
2668:
2639:
2619:
2593:
2548:
2524:, then the growth of
2519:
2499:
2473:
2431:
2399:
2373:
2321:
2276:
2244:
2138:
1935:
1764:
1647:Historical background
1622:
1561:
1521:
1501:
1499:{\displaystyle s_{i}}
1474:
1438:
1362:
1329:
1296:
1244:renormalization group
1169:Large Hadron Collider
294:KleinâGordon equation
236:LSZ reduction formula
3375:Mathematical physics
2950:
2784:
2769:reductio ad absurdum
2733:
2704:
2677:
2648:
2628:
2602:
2560:
2528:
2508:
2482:
2440:
2411:
2382:
2340:
2301:
2256:
2226:
2031:
1849:
1678:
1570:
1534:
1510:
1483:
1447:
1386:
1338:
1305:
1285:
1252:lattice gauge theory
1132:quantum field theory
377:Theory of everything
231:Lattice field theory
201:Correlation function
23:Quantum field theory
3313:1986ZPhyC..31..295L
3276:1984NuPhB.233..189C
3239:2010mfq..confE..39F
3177:2010JETP..111..450S
3122:1987NuPhB.292..497C
2923:1988PhR...167..241C
2336:is saturated, i.e.
356:Incomplete theories
3380:Physical phenomena
3321:10.1007/BF01479540
2963:
2797:
2752:
2719:
2690:
2663:
2634:
2614:
2588:
2543:
2514:
2494:
2468:
2426:
2394:
2368:
2316:
2271:
2239:
2133:
1969:invalidates Eqs. (
1930:
1759:
1617:
1556:
1516:
1496:
1469:
1433:
1369:partition function
1357:
1324:
1291:
1276:critical phenomena
1222:. You can help by
1141:quantum triviality
241:Partition function
168:Topological charge
88:General relativity
83:Special relativity
3370:Quantum mechanics
3264:Nuclear Physics B
3109:Nuclear Physics B
3088:978-0-471-04223-5
2905:D. J. E. Callaway
2844:Hierarchy problem
2637:{\displaystyle g}
2517:{\displaystyle g}
2332:, then growth of
2159:which gives Eqs.(
2157:
2156:
2129:
2058:
1954:
1953:
1926:
1922:
1901:
1874:
1783:
1782:
1755:
1751:
1688:
1605:
1547:
1519:{\displaystyle Z}
1460:
1421:
1294:{\displaystyle Z}
1240:
1239:
1197:asymptotic safety
1128:
1127:
221:Expectation value
196:BRST quantization
143:Poincaré symmetry
98:YangâMills theory
78:Quantum mechanics
3387:
3350:
3339:
3333:
3332:
3294:
3288:
3287:
3255:
3249:
3248:
3246:
3245:
3232:
3214:
3203:
3197:
3196:
3170:
3147:Function in the
3140:
3134:
3133:
3099:
3093:
3092:
3077:(3rd ed.).
3070:
3064:
3063:
3049:
3043:
3042:
3028:
3022:
3021:
3000:
2994:
2984:
2978:
2972:
2970:
2969:
2964:
2962:
2961:
2941:
2935:
2934:
2901:
2890:
2889:
2859:
2831:particle physics
2816:
2806:
2804:
2803:
2798:
2796:
2795:
2779:
2761:
2759:
2758:
2753:
2751:
2750:
2728:
2726:
2725:
2720:
2699:
2697:
2696:
2691:
2689:
2688:
2672:
2670:
2669:
2664:
2643:
2641:
2640:
2635:
2623:
2621:
2620:
2615:
2597:
2595:
2594:
2589:
2587:
2586:
2552:
2550:
2549:
2544:
2523:
2521:
2520:
2515:
2503:
2501:
2500:
2495:
2477:
2475:
2474:
2469:
2467:
2466:
2435:
2433:
2432:
2427:
2403:
2401:
2400:
2395:
2377:
2375:
2374:
2369:
2367:
2366:
2335:
2331:
2325:
2323:
2322:
2317:
2291:
2280:
2278:
2277:
2272:
2248:
2246:
2245:
2240:
2238:
2237:
2221:
2214:
2197:
2187:
2151:
2142:
2140:
2139:
2134:
2127:
2120:
2119:
2110:
2109:
2097:
2096:
2087:
2086:
2059:
2057:
2043:
2035:
2025:
2018:
2014:
2000:
1990:
1981:) in the region
1968:
1964:
1948:
1939:
1937:
1936:
1931:
1924:
1923:
1921:
1917:
1903:
1902:
1899:
1893:
1892:
1876:
1875:
1872:
1866:
1861:
1860:
1843:
1839:
1830:
1826:
1814:
1810:
1802:is finite, then
1801:
1792:
1788:
1777:
1768:
1766:
1765:
1760:
1753:
1752:
1750:
1746:
1732:
1731:
1722:
1721:
1705:
1704:
1695:
1690:
1689:
1686:
1672:
1668:
1659:
1626:
1624:
1623:
1618:
1613:
1612:
1607:
1606:
1598:
1585:
1584:
1565:
1563:
1562:
1557:
1555:
1554:
1549:
1548:
1540:
1530:in terms of the
1525:
1523:
1522:
1517:
1505:
1503:
1502:
1497:
1495:
1494:
1478:
1476:
1475:
1470:
1468:
1467:
1462:
1461:
1453:
1443:, the number of
1442:
1440:
1439:
1434:
1429:
1428:
1423:
1422:
1414:
1401:
1400:
1366:
1364:
1363:
1358:
1353:
1352:
1333:
1331:
1330:
1325:
1320:
1319:
1300:
1298:
1297:
1292:
1235:
1232:
1214:
1207:
1154:particle physics
1136:charge screening
1120:
1113:
1106:
211:Effective action
138:Lorentz symmetry
63:Electromagnetism
33:
19:
18:
3395:
3394:
3390:
3389:
3388:
3386:
3385:
3384:
3355:
3354:
3353:
3340:
3336:
3295:
3291:
3256:
3252:
3243:
3241:
3212:
3204:
3200:
3141:
3137:
3100:
3096:
3089:
3071:
3067:
3050:
3046:
3029:
3025:
3008:A. A. Abrikosov
3001:
2997:
2985:
2981:
2957:
2953:
2951:
2948:
2947:
2942:
2938:
2910:Physics Reports
2902:
2893:
2886:
2860:
2856:
2852:
2840:
2823:
2814:
2808:
2791:
2787:
2785:
2782:
2781:
2778:
2772:
2765:
2746:
2742:
2734:
2731:
2730:
2705:
2702:
2701:
2684:
2680:
2678:
2675:
2674:
2649:
2646:
2645:
2629:
2626:
2625:
2603:
2600:
2599:
2582:
2578:
2561:
2558:
2557:
2529:
2526:
2525:
2509:
2506:
2505:
2483:
2480:
2479:
2462:
2458:
2441:
2438:
2437:
2412:
2409:
2408:
2383:
2380:
2379:
2362:
2358:
2341:
2338:
2337:
2333:
2327:
2302:
2299:
2298:
2282:
2257:
2254:
2253:
2233:
2229:
2227:
2224:
2223:
2216:
2213:
2199:
2189:
2186:
2172:
2115:
2111:
2105:
2101:
2092:
2088:
2082:
2078:
2044:
2036:
2034:
2032:
2029:
2028:
2016:
2005:
1998:
1992:
1988:
1982:
1966:
1963:
1957:
1913:
1898:
1894:
1888:
1884:
1877:
1871:
1867:
1865:
1856:
1852:
1850:
1847:
1846:
1838:
1832:
1828:
1825:
1819:
1812:
1809:
1803:
1800:
1794:
1790:
1786:
1742:
1727:
1723:
1717:
1713:
1706:
1700:
1696:
1694:
1685:
1681:
1679:
1676:
1675:
1667:
1661:
1658:
1652:
1649:
1608:
1597:
1596:
1595:
1580:
1576:
1571:
1568:
1567:
1550:
1539:
1538:
1537:
1535:
1532:
1531:
1511:
1508:
1507:
1490:
1486:
1484:
1481:
1480:
1463:
1452:
1451:
1450:
1448:
1445:
1444:
1424:
1413:
1412:
1411:
1396:
1392:
1387:
1384:
1383:
1348:
1344:
1339:
1336:
1335:
1315:
1311:
1306:
1303:
1302:
1286:
1283:
1282:
1260:Leo P. Kadanoff
1236:
1230:
1227:
1220:needs expansion
1205:
1124:
1095:
1094:
1093:
1091:
395:
387:
386:
382:Quantum gravity
357:
349:
348:
344:Higgs mechanism
324:
314:
313:
299:Proca equations
284:
276:
275:
261:Renormalization
226:Feynman diagram
181:
173:
172:
113:
103:
102:
53:
38:
36:Feynman diagram
17:
12:
11:
5:
3393:
3383:
3382:
3377:
3372:
3367:
3352:
3351:
3334:
3307:(2): 295â300.
3289:
3270:(2): 189â203.
3250:
3198:
3161:(3): 450â465.
3135:
3094:
3087:
3065:
3044:
3023:
2995:
2979:
2960:
2956:
2936:
2917:(5): 241â320.
2891:
2884:
2862:R. Fernandez;
2853:
2851:
2848:
2847:
2846:
2839:
2836:
2827:Standard Model
2822:
2819:
2812:
2794:
2790:
2776:
2764:
2763:
2749:
2745:
2741:
2738:
2718:
2715:
2712:
2709:
2687:
2683:
2662:
2659:
2656:
2653:
2633:
2613:
2610:
2607:
2585:
2581:
2577:
2574:
2571:
2568:
2565:
2554:
2542:
2539:
2536:
2533:
2513:
2493:
2490:
2487:
2465:
2461:
2457:
2454:
2451:
2448:
2445:
2425:
2422:
2419:
2416:
2405:
2393:
2390:
2387:
2365:
2361:
2357:
2354:
2351:
2348:
2345:
2315:
2312:
2309:
2306:
2294:
2270:
2267:
2264:
2261:
2236:
2232:
2211:
2184:
2155:
2154:
2145:
2143:
2132:
2126:
2123:
2118:
2114:
2108:
2104:
2100:
2095:
2091:
2085:
2081:
2077:
2074:
2071:
2068:
2065:
2062:
2056:
2053:
2050:
2047:
2042:
2039:
1996:
1986:
1961:
1956:The growth of
1952:
1951:
1942:
1940:
1929:
1920:
1916:
1912:
1909:
1906:
1897:
1891:
1887:
1883:
1880:
1870:
1864:
1859:
1855:
1836:
1823:
1807:
1798:
1781:
1780:
1771:
1769:
1758:
1749:
1745:
1741:
1738:
1735:
1730:
1726:
1720:
1716:
1712:
1709:
1703:
1699:
1693:
1684:
1665:
1656:
1648:
1645:
1629:renormalizable
1616:
1611:
1604:
1601:
1594:
1591:
1588:
1583:
1579:
1575:
1553:
1546:
1543:
1515:
1493:
1489:
1466:
1459:
1456:
1432:
1427:
1420:
1417:
1410:
1407:
1404:
1399:
1395:
1391:
1356:
1351:
1347:
1343:
1323:
1318:
1314:
1310:
1290:
1256:renormalizable
1248:Kenneth Wilson
1238:
1237:
1217:
1215:
1204:
1201:
1177:W and Z bosons
1150:Standard Model
1126:
1125:
1123:
1122:
1115:
1108:
1100:
1097:
1096:
1089:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1053:
1048:
1043:
1038:
1033:
1028:
1023:
1018:
1013:
1008:
1003:
998:
993:
988:
983:
978:
973:
968:
963:
958:
953:
948:
943:
938:
933:
928:
923:
918:
913:
908:
903:
898:
893:
888:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
628:
623:
618:
613:
608:
603:
598:
593:
588:
583:
578:
573:
568:
563:
558:
553:
548:
543:
538:
533:
528:
523:
518:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
448:
443:
438:
433:
428:
423:
418:
413:
408:
403:
397:
396:
393:
392:
389:
388:
385:
384:
379:
374:
369:
364:
358:
355:
354:
351:
350:
347:
346:
341:
336:
331:
325:
322:Standard Model
320:
319:
316:
315:
312:
311:
306:
301:
296:
291:
289:Dirac equation
285:
282:
281:
278:
277:
274:
273:
271:Wick's theorem
268:
263:
258:
256:Regularization
253:
248:
243:
238:
233:
228:
223:
218:
213:
208:
203:
198:
193:
188:
182:
179:
178:
175:
174:
171:
170:
165:
163:Noether charge
160:
155:
150:
148:Gauge symmetry
145:
140:
135:
130:
125:
120:
114:
109:
108:
105:
104:
101:
100:
95:
90:
85:
80:
75:
70:
65:
60:
54:
51:
50:
47:
46:
40:
39:
34:
26:
25:
15:
9:
6:
4:
3:
2:
3392:
3381:
3378:
3376:
3373:
3371:
3368:
3366:
3363:
3362:
3360:
3348:
3344:
3338:
3330:
3326:
3322:
3318:
3314:
3310:
3306:
3302:
3301:
3293:
3285:
3281:
3277:
3273:
3269:
3265:
3261:
3254:
3240:
3236:
3231:
3226:
3222:
3218:
3211:
3210:
3202:
3194:
3190:
3186:
3182:
3178:
3174:
3169:
3164:
3160:
3156:
3155:
3150:
3146:
3139:
3131:
3127:
3123:
3119:
3115:
3111:
3110:
3105:
3098:
3090:
3084:
3080:
3076:
3069:
3061:
3057:
3056:
3048:
3040:
3036:
3035:
3027:
3019:
3015:
3014:
3009:
3005:
2999:
2992:
2988:
2983:
2976:
2958:
2954:
2945:
2944:L.P. Kadanoff
2940:
2932:
2928:
2924:
2920:
2916:
2912:
2911:
2906:
2900:
2898:
2896:
2887:
2885:0-387-54358-9
2881:
2877:
2873:
2869:
2865:
2858:
2854:
2845:
2842:
2841:
2835:
2832:
2828:
2818:
2811:
2792:
2788:
2775:
2771:. Indeed, if
2770:
2747:
2743:
2739:
2736:
2713:
2707:
2685:
2681:
2657:
2651:
2631:
2611:
2608:
2605:
2583:
2579:
2575:
2569:
2563:
2555:
2537:
2531:
2511:
2491:
2488:
2485:
2463:
2459:
2455:
2449:
2443:
2420:
2414:
2406:
2385:
2363:
2359:
2349:
2343:
2330:
2310:
2304:
2296:
2295:
2293:
2289:
2285:
2265:
2259:
2250:
2234:
2230:
2219:
2210:
2206:
2202:
2196:
2192:
2183:
2179:
2175:
2170:
2169:
2164:
2163:
2153:
2146:
2144:
2130:
2124:
2121:
2116:
2112:
2106:
2102:
2098:
2093:
2089:
2083:
2079:
2075:
2069:
2063:
2060:
2054:
2051:
2048:
2045:
2040:
2037:
2027:
2026:
2023:
2022:
2012:
2008:
2002:
1995:
1985:
1980:
1979:
1974:
1973:
1960:
1950:
1943:
1941:
1927:
1918:
1914:
1907:
1904:
1895:
1889:
1885:
1881:
1878:
1868:
1862:
1857:
1853:
1845:
1844:
1841:
1835:
1822:
1816:
1806:
1797:
1779:
1772:
1770:
1756:
1747:
1743:
1736:
1733:
1728:
1724:
1718:
1714:
1710:
1707:
1701:
1697:
1691:
1682:
1674:
1673:
1670:
1664:
1655:
1644:
1642:
1638:
1634:
1630:
1609:
1599:
1581:
1577:
1551:
1541:
1529:
1513:
1491:
1487:
1464:
1454:
1425:
1415:
1397:
1393:
1380:
1378:
1374:
1370:
1349:
1345:
1316:
1312:
1288:
1279:
1277:
1273:
1272:Kondo problem
1267:
1265:
1264:blocking idea
1261:
1257:
1253:
1249:
1245:
1234:
1225:
1221:
1218:This section
1216:
1213:
1209:
1208:
1200:
1198:
1192:
1190:
1186:
1182:
1179:, as well as
1178:
1174:
1170:
1166:
1162:
1157:
1155:
1151:
1146:
1142:
1137:
1133:
1121:
1116:
1114:
1109:
1107:
1102:
1101:
1099:
1098:
1092:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1061:Zamolodchikov
1059:
1057:
1056:Zamolodchikov
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1037:
1034:
1032:
1029:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
522:
519:
517:
514:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
454:
452:
449:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
422:
419:
417:
414:
412:
409:
407:
404:
402:
399:
398:
391:
390:
383:
380:
378:
375:
373:
370:
368:
367:Supersymmetry
365:
363:
362:String theory
360:
359:
353:
352:
345:
342:
340:
337:
335:
332:
330:
327:
326:
323:
318:
317:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
286:
280:
279:
272:
269:
267:
264:
262:
259:
257:
254:
252:
249:
247:
244:
242:
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
183:
177:
176:
169:
166:
164:
161:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
119:
116:
115:
112:
107:
106:
99:
96:
94:
91:
89:
86:
84:
81:
79:
76:
74:
71:
69:
66:
64:
61:
59:
56:
55:
49:
48:
45:
42:
41:
37:
32:
28:
27:
24:
21:
20:
3346:
3342:
3337:
3304:
3298:
3292:
3267:
3263:
3253:
3242:. Retrieved
3208:
3201:
3158:
3152:
3148:
3144:
3138:
3113:
3107:
3097:
3074:
3068:
3059:
3053:
3047:
3038:
3032:
3026:
3017:
3011:
3004:L. D. Landau
2998:
2990:
2982:
2974:
2939:
2914:
2908:
2871:
2864:J. Froehlich
2857:
2824:
2809:
2773:
2766:
2328:
2287:
2283:
2251:
2217:
2208:
2204:
2200:
2194:
2190:
2181:
2177:
2173:
2166:
2160:
2158:
2147:
2010:
2006:
2003:
1993:
1983:
1976:
1970:
1958:
1955:
1944:
1833:
1820:
1817:
1804:
1795:
1784:
1773:
1662:
1653:
1650:
1636:
1633:fixed points
1632:
1628:
1527:
1381:
1280:
1268:
1263:
1255:
1241:
1228:
1224:adding to it
1219:
1193:
1158:
1140:
1129:
1090:
936:Stueckelberg
676:Jona-Lasinio
266:Vacuum state
251:Quantization
93:Gauge theory
73:Strong force
58:Field theory
3343:Nucl. Phys.
3116:: 497â526.
2987:K.G. Wilson
2868:A. D. Sokal
2821:Conclusions
1377:Hamiltonian
1199:scenarios.
1163:problem in
1161:Landau pole
1145:Higgs boson
1076:Zinn-Justin
926:Sommerfield
851:Pomeranchuk
821:Osterwalder
816:Oppenheimer
746:ĆopuszaĆski
571:Fredenhagen
372:Technicolor
3359:Categories
3349:: 555-573.
3244:2011-08-27
2850:References
2624:for large
2504:for large
1071:Zimmermann
966:Vainshtein
711:Kontsevich
656:Iliopoulos
631:Heisenberg
456:Bogoliubov
394:Scientists
246:Propagator
133:T-symmetry
128:P-symmetry
123:C-symmetry
111:Symmetries
68:Weak force
52:Background
3329:123166350
3230:1011.3643
3193:118545858
3168:1010.4317
2993:, 4, 773.
2789:μ
2744:μ
2737:μ
2714:μ
2682:μ
2658:μ
2606:α
2584:α
2576:∝
2564:β
2538:μ
2489:≤
2486:α
2464:α
2456:∝
2444:β
2415:β
2392:∞
2389:→
2386:μ
2364:∗
2356:→
2350:μ
2305:β
2266:μ
2231:β
2125:…
2103:β
2080:β
2064:β
2055:μ
2052:
1911:Λ
1908:
1886:β
1882:−
1740:Λ
1737:
1715:β
1603:~
1590:→
1545:~
1526:function
1458:~
1419:~
1406:→
1231:July 2019
1171:(LHC) at
1006:Wetterich
991:Weisskopf
941:Sudarshan
891:Schwinger
806:Nishijima
771:Maldacena
736:Leutwyler
701:Kinoshita
601:Goldstone
591:Gell-Mann
506:Doplicher
283:Equations
2876:Springer
2870:(1992).
2838:See also
1185:electron
1021:Wightman
986:Weinberg
976:Virasoro
956:Tomonaga
951:Thirring
946:Symanzik
906:Semenoff
881:Schrader
846:Polyakov
766:Majorana
706:Klebanov
661:Ivanenko
651:'t Hooft
621:Guralnik
566:Fröhlich
561:Fritzsch
556:Frampton
471:Buchholz
416:Bargmann
406:Anderson
206:Crossing
3309:Bibcode
3272:Bibcode
3235:Bibcode
3221:Trieste
3173:Bibcode
3118:Bibcode
3062:: 1177.
2919:Bibcode
2644:, then
1975:) and (
1637:trivial
1031:Wilczek
996:Wentzel
971:Veltman
916:Shirkov
911:Shifman
901:Seiberg
886:Schwarz
866:Rubakov
791:Naimark
741:Lipatov
731:Lehmann
696:Kendall
586:Gelfand
581:Glashow
541:Feynman
521:Faddeev
516:Englert
486:Coleman
476:Cachazo
461:Brodsky
446:Bjorken
436:Berezin
426:Belavin
186:Anomaly
44:History
3327:
3191:
3085:
3041:: 773.
3020:: 497.
2977:, 263.
2882:
2128:
1925:
1785:where
1754:
1373:action
1181:lepton
1086:Zumino
1051:Yukawa
1041:Witten
1036:Wilson
1026:Wigner
961:Tyutin
921:Skyrme
871:Ruelle
841:Plefka
836:Peskin
826:Parisi
786:MĂžller
776:Migdal
761:Maiani
756:LĂŒders
721:Landau
716:Kuraev
691:KÀllén
681:Jordan
666:Jackiw
606:Gribov
496:DeWitt
491:Dashen
481:Callan
451:Bleuer
421:Becchi
411:Anselm
3325:S2CID
3225:arXiv
3213:(PDF)
3189:S2CID
3163:arXiv
2598:with
2478:with
1965:with
1371:, an
1130:In a
1081:Zuber
931:Stora
896:Segal
876:Salam
861:Proca
856:Popov
831:Pauli
811:Oehme
801:Neveu
796:Nambu
781:Mills
671:Jaffe
646:Hagen
641:Higgs
616:Gupta
611:Gross
596:Glimm
576:Furry
546:Fierz
536:Fermi
531:Fayet
526:Fadin
511:Dyson
501:Dirac
466:Brout
441:Bethe
401:Adler
180:Tools
3347:B405
3083:ISBN
2880:ISBN
2740:>
2729:for
2609:>
2378:for
2215:for
2207:) =
2198:and
2188:for
2180:) =
1528:only
1375:, a
1189:muon
1187:and
1173:CERN
1046:Yang
1016:Wick
1011:Weyl
1001:Wess
981:Ward
686:Jost
636:Hepp
626:Haag
551:Fock
431:Bell
3317:doi
3280:doi
3268:233
3181:doi
3159:111
3126:doi
3114:292
2927:doi
2915:167
2829:of
2815:â 0
2813:obs
2777:obs
2556:if
2407:if
2297:if
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2185:obs
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1999:âȘ 1
1989:â 1
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1873:obs
1837:obs
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1808:obs
1687:obs
1669:,
1657:obs
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726:Lee
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2878:.
2874:.
2866:;
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2193:=
2049:ln
1905:ln
1840:,
1815:.
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2178:Ό
2176:(
2174:g
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2162:1
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2150:3
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2122:+
2117:3
2113:g
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2099:+
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2070:g
2067:(
2061:=
2046:d
2041:g
2038:d
2017:Ό
2013:)
2011:Ό
2009:(
2007:g
1997:0
1994:g
1987:0
1984:g
1978:2
1972:1
1967:Î
1962:0
1959:g
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1947:2
1945:(
1928:.
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1915:/
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1879:1
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1821:g
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1729:0
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1342:{
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1105:v
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