2255:
1414:
2250:{\displaystyle {\begin{aligned}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta _{\mathrm {CP} }}\\0&1&0\\-s_{13}e^{i\delta _{\mathrm {CP} }}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{\mathrm {CP} }}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&c_{23}c_{13}\end{bmatrix}},\end{aligned}}}
1117:
2768:
761:
2490:
3263:
1112:{\displaystyle {\begin{bmatrix}~\nu _{\mathrm {e} }\\~\nu _{\mu }\\~\nu _{\tau }~\end{bmatrix}}={\begin{bmatrix}~U_{\mathrm {e} 1}~&~U_{\mathrm {e} 2}~&~U_{\mathrm {e} 3}\\~U_{\mu 1}&~U_{\mu 2}~&~U_{\mu 3}\\~U_{\tau 1}~&~U_{\tau 2}~&~U_{\tau 3}\end{bmatrix}}{\begin{bmatrix}~\nu _{1}\\~\nu _{2}\\~\nu _{3}~\end{bmatrix}}~.}
2763:{\displaystyle {\begin{aligned}\theta _{12}&={33.41^{\circ }}_{-0.72^{\circ }}^{+0.75^{\circ }}\\\theta _{23}&={49.1^{\circ }}_{-1.3^{\circ }}^{+1.0^{\circ }}\\\theta _{13}&={8.54^{\circ }}_{-0.12^{\circ }}^{+0.11^{\circ }}\\\delta _{\textrm {CP}}&={197^{\circ }}_{-25^{\circ }}^{+42^{\circ }}\\\end{aligned}}}
1279:
There are also additional mass parameters and mixing angles in a simple extension of the PMNS matrix in which there are more than three flavors of neutrinos, regardless of the character of neutrino mass. As of July 2014, scientists studying neutrino oscillation are actively considering fits of
1288:
In general, there are nine degrees of freedom in any unitary three by three matrix. However, in the case of the PMNS matrix, five of those real parameters can be absorbed as phases of the lepton fields and thus the PMNS matrix can be fully described by four free parameters. The PMNS matrix is most
2782:
1267:
In other models the PMNS matrix is not necessarily unitary, and additional parameters are necessary to describe all possible neutrino mixing parameters in other models of neutrino oscillation and mass generation, such as the see-saw model, and in general, in the case of neutrinos that have
1122:
The vector on the left represents a generic neutrino expressed in the flavor-eigenstate basis, and on the right is the PMNS matrix multiplied by a vector representing that same neutrino in the mass-eigenstate basis. A neutrino of a given flavor
1406:(i.e. differences in the rates of oscillation between two states with opposite starting points which makes the order in time in which events take place necessary to predict their oscillation rates), in which case the matrix can be written as:
1258:
In the simplest case, the
Standard Model posits three generations of neutrinos with Dirac mass that oscillate between three neutrino mass eigenvalues, an assumption that is made when best fit values for its parameters are calculated.
3529:) at a statistical significance of more than five standard deviations. Tribimaximal neutrino mixing was a common assumption in theoretical physics papers analyzing neutrino oscillation before more precise measurements were available.
3606:
3258:{\displaystyle |U|={\begin{bmatrix}~|U_{\mathrm {e} 1}|~&|U_{\mathrm {e} 2}|~&|U_{\mathrm {e} 3}|\\~|U_{\mu 1}|~&|U_{\mu 2}|~&|U_{\mu 3}|\\~|U_{\tau 1}|~&|U_{\tau 2}|~&|U_{\tau 3}|~\end{bmatrix}}=\left}
3667:
1280:
the experimental neutrino oscillation data to an extended PMNS matrix with a fourth, light "sterile" neutrino and four mass eigenvalues, although the current experimental data tends to disfavor that possibility.
1255:. This implies that the sum of the squares of the values in each row and in each column, which represent the probabilities of different possible events given the same starting point, add up to 100%.
3781:
3444:
2495:
1419:
1214:
3275:
These best fit values imply that there is much more neutrino mixing than there is mixing between the quark flavors in the CKM matrix (in the CKM matrix, the corresponding mixing angles are
3487:
3527:
2474:
1400:
473:
2392:
2356:
3385:
3345:
3305:
2429:
674:
1368:
1341:
1314:
527:
741:
714:
500:
340:
4075:
Gonzalez-Garcia, M.C.; Maltoni, Michele; Salvado, Jordi; Schwetz, Thomas (21 December 2012). "Global fit to three neutrino mixing: Critical look at present precision".
628:
601:
574:
2394:
respectively. In the case of
Majorana neutrinos, two extra complex phases are needed, as the phase of Majorana fields cannot be freely redefined due to the condition
1141:
2320:
2290:
1168:
3535:
3689:
2434:
1235:
304:
3877:
Esmaili, Arman; Kemp, Ernesto; Peres, O.L.G.; Tabrizi, Zahra (30 Oct 2013). "Probing light sterile neutrinos in medium baseline reactor experiments".
3611:
3995:
333:
1143:
is thus a "mixed" state of neutrinos with distinct mass: If one could measure directly that neutrino's mass, it would be found to have mass
2484:
As of
November 2022, the current best-fit values from Nu-FIT.org, from direct and indirect measurements, using normal ordering, are:
314:
326:
3742:
631:
4077:
3405:
439:
1234:, it is much more difficult to determine the individual coefficients than in the equivalent matrix for the quarks (the
1173:
33:
3932:
An, F.P.; et al. (Daya Bay collaboration) (27 July 2014). "Search for a light sterile neutrino at Daya Bay".
3448:
3491:
2447:
1373:
4152:
4104:
3841:
71:
4051:
Esteban, Ivan; Gonzalez Garcia, Concha; Maltoni, Michele; Schwetz, Thomas; Albert, Zhou (November 2022).
449:
642:
2361:
2325:
4096:
4014:
3951:
3898:
3833:
3751:
3679:
3608:
is very difficult to measure, and is the object of ongoing research; however the current constraint
3359:
3319:
3279:
415:
407:
4109:
3846:
2476:
has not been measured directly, but estimates can be obtained by fits using the other measurements.
2398:
648:
641:
Consequently, each flavor eigenstate can be written as a combination of mass eigenstates, called a "
3795:
3399:
1252:
1346:
1319:
1292:
505:
4142:
4122:
4086:
4030:
4004:
3975:
3941:
3914:
3888:
3859:
3823:
719:
679:
478:
411:
3967:
3879:
1231:
606:
579:
552:
542:
443:
1126:
634:. Observations of neutrino oscillation established experimentally that for neutrinos, as for
4147:
4114:
4022:
3959:
3906:
3851:
3759:
3713:
2295:
2265:
1269:
534:
423:
403:
354:
236:
38:
2433:
An infinite number of possible parameterizations exist; one other common example being the
1146:
17:
2441:
153:
4100:
4018:
3955:
3902:
3837:
3755:
435:
419:
388:
46:
3779:
Pontecorvo, B. (1957). "Inverse beta processes and nonconservation of lepton charge".
3089:
4136:
4126:
4026:
3918:
3863:
3684:
1273:
395:
128:
119:
3979:
4034:
3963:
3601:{\displaystyle \delta _{\textrm {CP}}={197^{\circ }}_{-25^{\circ }}^{+42^{\circ }}}
1403:
1224:
1220:
391:
137:
638:, these two eigenbases are different – they are 'rotated' relative to each other.
2777:
ranges (99.7% confidence) for the magnitudes of the elements of the matrix were:
4118:
3662:{\displaystyle \,{169^{\circ }}\leq \delta _{\textrm {CP}}\leq {246^{\circ }}\,}
538:
182:
80:
3910:
3669:
in the vicinity of 180° shows a clear bias in favor of charge-parity violation.
546:
162:
98:
279:
3971:
399:
382:
4009:
3764:
3737:
89:
55:
3855:
2440:
The mixing angles have been measured by a variety of experiments (see
635:
530:
375:
4052:
755:"; parameterizes the unitary transformation between the two bases:
4091:
3946:
3893:
3828:
3814:
Kayser, Boris (13 February 2014). "Are there sterile neutrinos?".
545:
for the
Standard Model neutrino. Similarly, one can construct an
3738:"Remarks on the unified model of elementary particles"
2807:
1773:
1669:
1522:
1428:
1042:
841:
770:
402:
when they propagate freely and when they take part in
3614:
3538:
3494:
3451:
3408:
3362:
3322:
3282:
2785:
2493:
2450:
2401:
2364:
2328:
2298:
2268:
1417:
1376:
1349:
1322:
1295:
1176:
1149:
1129:
764:
722:
682:
651:
609:
582:
555:
508:
481:
452:
3439:{\displaystyle \theta _{12}\approx 35.3^{\circ }\,,}
645:", and vice versa. The PMNS matrix, with components
529:, each labeled with a subscript showing the charged
422:, to explain the neutrino oscillations predicted by
3993:Valle, J.W.F. (2006). "Neutrino physics overview".
3661:
3600:
3521:
3481:
3438:
3379:
3339:
3299:
3257:
2762:
2468:
2423:
2386:
2350:
2314:
2284:
2249:
1394:
1362:
1335:
1308:
1208:
1162:
1135:
1111:
735:
708:
676:corresponding to the amplitude of mass eigenstate
668:
622:
595:
568:
521:
494:
467:
3782:Zhurnal Éksperimental'noĭ i Teoreticheskoĭ Fiziki
630:, which diagonalize the neutrino's free-particle
1289:commonly parameterized by three mixing angles (
1223:is identical to the matrix for neutrinos under
1209:{\displaystyle \left|U_{\alpha \,i}\right|^{2}}
549:out of three neutrino states of definite mass,
394:which contains information on the mismatch of
3268:Notes regarding the best fit parameter values
334:
8:
3736:Maki, Z.; Nakagawa, M.; Sakata, S. (1962).
3482:{\displaystyle \theta _{23}=45^{\circ }\,,}
2444:for a description). The CP-violating phase
1251:In the Standard Model, the PMNS matrix is
732:
705:
341:
327:
29:
4108:
4090:
4059:. Three-neutrino fit (NuFIT 5.2 ed.)
4008:
3945:
3892:
3845:
3827:
3763:
3658:
3651:
3646:
3636:
3635:
3621:
3616:
3615:
3613:
3590:
3582:
3575:
3567:
3560:
3555:
3544:
3543:
3537:
3522:{\displaystyle \theta _{13}=0^{\circ }\,}
3518:
3512:
3499:
3493:
3475:
3469:
3456:
3450:
3432:
3426:
3413:
3407:
3376:
3367:
3361:
3336:
3327:
3321:
3296:
3287:
3281:
3088:
3065:
3056:
3047:
3037:
3028:
3019:
3009:
3000:
2991:
2979:
2970:
2961:
2951:
2942:
2933:
2923:
2914:
2905:
2893:
2883:
2882:
2873:
2863:
2853:
2852:
2843:
2833:
2823:
2822:
2813:
2802:
2794:
2786:
2784:
2748:
2740:
2733:
2725:
2718:
2713:
2698:
2697:
2681:
2673:
2666:
2658:
2651:
2646:
2632:
2616:
2608:
2601:
2593:
2586:
2581:
2567:
2551:
2543:
2536:
2528:
2521:
2516:
2502:
2494:
2492:
2456:
2455:
2449:
2412:
2400:
2375:
2363:
2339:
2327:
2303:
2297:
2273:
2267:
2226:
2216:
2198:
2197:
2189:
2179:
2169:
2159:
2146:
2136:
2115:
2114:
2106:
2096:
2086:
2076:
2063:
2053:
2039:
2029:
2011:
2010:
2002:
1992:
1982:
1972:
1959:
1949:
1931:
1930:
1922:
1912:
1902:
1892:
1879:
1869:
1846:
1845:
1834:
1824:
1812:
1802:
1790:
1780:
1768:
1722:
1710:
1688:
1676:
1664:
1650:
1627:
1626:
1618:
1608:
1568:
1567:
1556:
1546:
1529:
1517:
1503:
1491:
1469:
1457:
1423:
1418:
1416:
1382:
1381:
1375:
1354:
1348:
1327:
1321:
1300:
1294:
1200:
1190:
1186:
1175:
1154:
1148:
1128:
1086:
1069:
1052:
1037:
1020:
999:
978:
958:
937:
919:
898:
897:
875:
874:
852:
851:
836:
816:
799:
781:
780:
765:
763:
721:
683:
681:
660:
656:
650:
614:
608:
587:
581:
560:
554:
541:of the weak interaction form a complete,
513:
507:
486:
480:
458:
457:
451:
2480:Experimentally measured parameter values
410:. This matrix was introduced in 1962 by
4046:
4044:
3728:
3701:
2469:{\displaystyle \delta _{\mathrm {CP} }}
1395:{\displaystyle \delta _{\mathrm {CP} }}
296:
174:
111:
44:
32:
3708:Note however, that the PMNS matrix is
359:Pontecorvo–Maki–Nakagawa–Sakata matrix
3996:Journal of Physics: Conference Series
2773:As of November 2022, the 3
7:
3398:These values are inconsistent with
468:{\displaystyle \nu _{\mathrm {e} }}
438:of particle physics contains three
2884:
2854:
2824:
2460:
2457:
2202:
2199:
2119:
2116:
2015:
2012:
1935:
1932:
1850:
1847:
1631:
1628:
1572:
1569:
1386:
1383:
1370:) and a single phase angle called
899:
876:
853:
782:
459:
25:
2387:{\displaystyle \cos \theta _{ij}}
2351:{\displaystyle \sin \theta _{ij}}
3690:Cabibbo–Kobayashi–Maskawa matrix
535:charged-current weak interaction
3743:Progress of Theoretical Physics
3380:{\displaystyle \theta _{13}=\,}
3340:{\displaystyle \theta _{23}=\,}
3300:{\displaystyle \theta _{12}=\,}
4078:Journal of High Energy Physics
3964:10.1103/PhysRevLett.113.141802
3818:. AIP Conference Proceedings.
3066:
3048:
3038:
3020:
3010:
2992:
2980:
2962:
2952:
2934:
2924:
2906:
2894:
2874:
2864:
2844:
2834:
2814:
2795:
2787:
2424:{\displaystyle \nu =\nu ^{c}~}
669:{\displaystyle U_{\alpha \,i}}
1:
3791:reproduced and translated in
533:that it partners with in the
27:Model of neutrino oscillation
3400:tribimaximal neutrino mixing
2435:Wolfenstein parameterization
1363:{\displaystyle \theta _{13}}
1336:{\displaystyle \theta _{23}}
1309:{\displaystyle \theta _{12}}
522:{\displaystyle \nu _{\tau }}
3793:Pontecorvo, B. (1958). "".
1230:Due to the difficulties of
736:{\displaystyle ~\alpha =\;}
709:{\displaystyle \,i=1,2,3\;}
495:{\displaystyle \nu _{\mu }}
367:Maki–Nakagawa–Sakata matrix
18:Maki-Nakagawa-Sakata matrix
4169:
4027:10.1088/1742-6596/53/1/031
3911:10.1103/PhysRevD.88.073012
1404:charge-parity violations
623:{\displaystyle \nu _{3}}
596:{\displaystyle \nu _{2}}
569:{\displaystyle \nu _{1}}
4119:10.1007/JHEP12(2012)123
3934:Physical Review Letters
1136:{\displaystyle \alpha }
315:Flavour complementarity
112:Related quantum numbers
3663:
3602:
3523:
3483:
3440:
3381:
3341:
3301:
3259:
2764:
2470:
2425:
2388:
2352:
2316:
2315:{\displaystyle c_{ij}}
2286:
2285:{\displaystyle s_{ij}}
2251:
1396:
1364:
1337:
1310:
1210:
1164:
1137:
1113:
737:
710:
670:
624:
597:
570:
523:
496:
469:
3664:
3603:
3524:
3484:
3441:
3382:
3342:
3302:
3260:
2765:
2471:
2426:
2389:
2353:
2317:
2287:
2252:
1397:
1365:
1338:
1311:
1211:
1165:
1163:{\displaystyle m_{i}}
1138:
1114:
738:
711:
671:
625:
598:
571:
524:
497:
470:
3680:Neutrino oscillation
3612:
3536:
3492:
3449:
3406:
3360:
3320:
3280:
2783:
2491:
2448:
2399:
2362:
2326:
2296:
2266:
1415:
1374:
1347:
1320:
1293:
1219:The PMNS matrix for
1174:
1147:
1127:
762:
720:
680:
649:
607:
580:
553:
506:
479:
450:
408:neutrino oscillation
4101:2012JHEP...12..123G
4019:2006JPhCS..53..473V
3956:2014PhRvL.113n1802A
3903:2013PhRvD..88g3012E
3838:2014AIPC.1604..201K
3796:Soviet Physics JETP
3756:1962PThPh..28..870M
3597:
2755:
2688:
2623:
2558:
2322:are used to denote
1232:detecting neutrinos
716:in terms of flavor
406:. It is a model of
4053:"Parameter ranges"
3765:10.1143/PTP.28.870
3659:
3598:
3554:
3519:
3479:
3436:
3377:
3337:
3297:
3255:
3249:
3075:
2760:
2758:
2712:
2645:
2580:
2515:
2466:
2421:
2384:
2348:
2312:
2282:
2247:
2245:
2234:
1752:
1658:
1511:
1392:
1360:
1333:
1306:
1206:
1160:
1133:
1109:
1097:
1031:
827:
733:
706:
666:
620:
593:
566:
519:
492:
465:
3880:Physical Review D
3856:10.1063/1.4883431
3639:
3547:
3246:
3232:
3229:
3215:
3212:
3200:
3193:
3179:
3176:
3162:
3159:
3147:
3140:
3126:
3123:
3109:
3106:
3094:
3072:
3044:
3016:
2990:
2958:
2930:
2904:
2870:
2840:
2812:
2701:
2420:
1170:with probability
1105:
1094:
1081:
1064:
1047:
1015:
1010:
994:
989:
973:
953:
948:
932:
914:
892:
887:
869:
864:
846:
824:
811:
794:
775:
725:
543:orthonormal basis
404:weak interactions
351:
350:
16:(Redirected from
4160:
4130:
4112:
4094:
4069:
4068:
4066:
4064:
4048:
4039:
4038:
4012:
3990:
3984:
3983:
3949:
3929:
3923:
3922:
3896:
3874:
3868:
3867:
3849:
3831:
3811:
3805:
3804:
3790:
3776:
3770:
3769:
3767:
3733:
3717:
3706:
3668:
3666:
3665:
3660:
3657:
3656:
3655:
3642:
3641:
3640:
3637:
3627:
3626:
3625:
3607:
3605:
3604:
3599:
3596:
3595:
3594:
3581:
3580:
3579:
3566:
3565:
3564:
3550:
3549:
3548:
3545:
3528:
3526:
3525:
3520:
3517:
3516:
3504:
3503:
3488:
3486:
3485:
3480:
3474:
3473:
3461:
3460:
3445:
3443:
3442:
3437:
3431:
3430:
3418:
3417:
3394:
3393:
3391:
3386:
3384:
3383:
3378:
3372:
3371:
3355:
3353:
3351:
3346:
3344:
3343:
3338:
3332:
3331:
3315:
3313:
3311:
3306:
3304:
3303:
3298:
3292:
3291:
3264:
3262:
3261:
3256:
3254:
3250:
3244:
3230:
3227:
3213:
3210:
3198:
3191:
3177:
3174:
3160:
3157:
3145:
3138:
3124:
3121:
3107:
3104:
3092:
3080:
3079:
3070:
3069:
3064:
3063:
3051:
3042:
3041:
3036:
3035:
3023:
3014:
3013:
3008:
3007:
2995:
2988:
2983:
2978:
2977:
2965:
2956:
2955:
2950:
2949:
2937:
2928:
2927:
2922:
2921:
2909:
2902:
2897:
2892:
2891:
2887:
2877:
2868:
2867:
2862:
2861:
2857:
2847:
2838:
2837:
2832:
2831:
2827:
2817:
2810:
2798:
2790:
2776:
2769:
2767:
2766:
2761:
2759:
2754:
2753:
2752:
2739:
2738:
2737:
2724:
2723:
2722:
2704:
2703:
2702:
2699:
2687:
2686:
2685:
2672:
2671:
2670:
2657:
2656:
2655:
2637:
2636:
2622:
2621:
2620:
2607:
2606:
2605:
2592:
2591:
2590:
2572:
2571:
2557:
2556:
2555:
2542:
2541:
2540:
2527:
2526:
2525:
2507:
2506:
2475:
2473:
2472:
2467:
2465:
2464:
2463:
2432:
2430:
2428:
2427:
2422:
2418:
2417:
2416:
2393:
2391:
2390:
2385:
2383:
2382:
2357:
2355:
2354:
2349:
2347:
2346:
2321:
2319:
2318:
2313:
2311:
2310:
2291:
2289:
2288:
2283:
2281:
2280:
2256:
2254:
2253:
2248:
2246:
2239:
2238:
2231:
2230:
2221:
2220:
2209:
2208:
2207:
2206:
2205:
2184:
2183:
2174:
2173:
2164:
2163:
2151:
2150:
2141:
2140:
2126:
2125:
2124:
2123:
2122:
2101:
2100:
2091:
2090:
2081:
2080:
2068:
2067:
2058:
2057:
2044:
2043:
2034:
2033:
2022:
2021:
2020:
2019:
2018:
1997:
1996:
1987:
1986:
1977:
1976:
1964:
1963:
1954:
1953:
1942:
1941:
1940:
1939:
1938:
1917:
1916:
1907:
1906:
1897:
1896:
1884:
1883:
1874:
1873:
1857:
1856:
1855:
1854:
1853:
1829:
1828:
1817:
1816:
1807:
1806:
1795:
1794:
1785:
1784:
1761:
1757:
1756:
1727:
1726:
1715:
1714:
1693:
1692:
1681:
1680:
1663:
1662:
1655:
1654:
1638:
1637:
1636:
1635:
1634:
1613:
1612:
1579:
1578:
1577:
1576:
1575:
1551:
1550:
1534:
1533:
1516:
1515:
1508:
1507:
1496:
1495:
1474:
1473:
1462:
1461:
1421:
1401:
1399:
1398:
1393:
1391:
1390:
1389:
1369:
1367:
1366:
1361:
1359:
1358:
1342:
1340:
1339:
1334:
1332:
1331:
1315:
1313:
1312:
1307:
1305:
1304:
1284:Parameterization
1215:
1213:
1212:
1207:
1205:
1204:
1199:
1195:
1194:
1169:
1167:
1166:
1161:
1159:
1158:
1142:
1140:
1139:
1134:
1118:
1116:
1115:
1110:
1103:
1102:
1101:
1092:
1091:
1090:
1079:
1074:
1073:
1062:
1057:
1056:
1045:
1036:
1035:
1028:
1027:
1013:
1008:
1007:
1006:
992:
987:
986:
985:
971:
966:
965:
951:
946:
945:
944:
930:
927:
926:
912:
907:
906:
902:
890:
885:
884:
883:
879:
867:
862:
861:
860:
856:
844:
832:
831:
822:
821:
820:
809:
804:
803:
792:
787:
786:
785:
773:
754:
750:
746:
742:
740:
739:
734:
723:
715:
713:
712:
707:
675:
673:
672:
667:
665:
664:
629:
627:
626:
621:
619:
618:
602:
600:
599:
594:
592:
591:
575:
573:
572:
567:
565:
564:
528:
526:
525:
520:
518:
517:
501:
499:
498:
493:
491:
490:
474:
472:
471:
466:
464:
463:
462:
446:" of neutrinos,
424:Bruno Pontecorvo
355:particle physics
343:
336:
329:
237:Weak hypercharge
39:particle physics
30:
21:
4168:
4167:
4163:
4162:
4161:
4159:
4158:
4157:
4133:
4132:
4110:10.1.1.762.7366
4074:
4072:
4062:
4060:
4050:
4049:
4042:
3992:
3991:
3987:
3931:
3930:
3926:
3876:
3875:
3871:
3847:10.1.1.761.2915
3813:
3812:
3808:
3792:
3778:
3777:
3773:
3735:
3734:
3730:
3726:
3721:
3720:
3712:unitary in the
3707:
3703:
3698:
3676:
3647:
3631:
3617:
3610:
3609:
3586:
3571:
3556:
3539:
3534:
3533:
3508:
3495:
3490:
3489:
3465:
3452:
3447:
3446:
3422:
3409:
3404:
3403:
3389:
3387:
3363:
3358:
3357:
3356:
3349:
3347:
3323:
3318:
3317:
3316:
3309:
3307:
3283:
3278:
3277:
3276:
3248:
3247:
3233:
3216:
3195:
3194:
3180:
3163:
3142:
3141:
3127:
3110:
3084:
3074:
3073:
3052:
3045:
3024:
3017:
2996:
2985:
2984:
2966:
2959:
2938:
2931:
2910:
2899:
2898:
2878:
2871:
2848:
2841:
2818:
2803:
2781:
2780:
2774:
2757:
2756:
2744:
2729:
2714:
2705:
2693:
2690:
2689:
2677:
2662:
2647:
2638:
2628:
2625:
2624:
2612:
2597:
2582:
2573:
2563:
2560:
2559:
2547:
2532:
2517:
2508:
2498:
2489:
2488:
2482:
2451:
2446:
2445:
2442:neutrino mixing
2408:
2397:
2396:
2395:
2371:
2360:
2359:
2335:
2324:
2323:
2299:
2294:
2293:
2269:
2264:
2263:
2244:
2243:
2233:
2232:
2222:
2212:
2210:
2193:
2185:
2175:
2165:
2155:
2142:
2132:
2127:
2110:
2102:
2092:
2082:
2072:
2059:
2049:
2046:
2045:
2035:
2025:
2023:
2006:
1998:
1988:
1978:
1968:
1955:
1945:
1943:
1926:
1918:
1908:
1898:
1888:
1875:
1865:
1859:
1858:
1841:
1830:
1820:
1818:
1808:
1798:
1796:
1786:
1776:
1769:
1759:
1758:
1751:
1750:
1745:
1740:
1734:
1733:
1728:
1718:
1716:
1706:
1700:
1699:
1694:
1684:
1682:
1672:
1665:
1657:
1656:
1646:
1644:
1639:
1622:
1614:
1604:
1598:
1597:
1592:
1587:
1581:
1580:
1563:
1552:
1542:
1540:
1535:
1525:
1518:
1510:
1509:
1499:
1497:
1487:
1482:
1476:
1475:
1465:
1463:
1453:
1451:
1445:
1444:
1439:
1434:
1424:
1413:
1412:
1377:
1372:
1371:
1350:
1345:
1344:
1323:
1318:
1317:
1296:
1291:
1290:
1286:
1265:
1249:
1244:
1182:
1178:
1177:
1172:
1171:
1150:
1145:
1144:
1125:
1124:
1096:
1095:
1082:
1076:
1075:
1065:
1059:
1058:
1048:
1038:
1030:
1029:
1016:
1011:
995:
990:
974:
968:
967:
954:
949:
933:
928:
915:
909:
908:
893:
888:
870:
865:
847:
837:
826:
825:
812:
806:
805:
795:
789:
788:
776:
766:
760:
759:
752:
748:
744:
718:
717:
678:
677:
652:
647:
646:
610:
605:
604:
583:
578:
577:
556:
551:
550:
509:
504:
503:
482:
477:
476:
453:
448:
447:
432:
430:The PMNS matrix
416:Masami Nakagawa
347:
277:
264:
253:
245:
230:
154:Electric charge
150:
68:
47:quantum numbers
37:
28:
23:
22:
15:
12:
11:
5:
4166:
4164:
4156:
4155:
4153:Standard Model
4150:
4145:
4135:
4134:
4071:
4070:
4040:
4010:hep-ph/0608101
4003:(1): 473–505.
3985:
3940:(14): 141802.
3924:
3869:
3822:(1): 201–203.
3806:
3771:
3727:
3725:
3722:
3719:
3718:
3700:
3699:
3697:
3694:
3693:
3692:
3687:
3682:
3675:
3672:
3671:
3670:
3654:
3650:
3645:
3634:
3630:
3624:
3620:
3593:
3589:
3585:
3578:
3574:
3570:
3563:
3559:
3553:
3542:
3530:
3515:
3511:
3507:
3502:
3498:
3478:
3472:
3468:
3464:
3459:
3455:
3435:
3429:
3425:
3421:
3416:
3412:
3396:
3375:
3370:
3366:
3335:
3330:
3326:
3295:
3290:
3286:
3272:
3271:
3269:
3253:
3243:
3240:
3237:
3234:
3226:
3223:
3220:
3217:
3209:
3206:
3203:
3197:
3196:
3190:
3187:
3184:
3181:
3173:
3170:
3167:
3164:
3156:
3153:
3150:
3144:
3143:
3137:
3134:
3131:
3128:
3120:
3117:
3114:
3111:
3103:
3100:
3097:
3091:
3090:
3087:
3083:
3078:
3068:
3062:
3059:
3055:
3050:
3046:
3040:
3034:
3031:
3027:
3022:
3018:
3012:
3006:
3003:
2999:
2994:
2987:
2986:
2982:
2976:
2973:
2969:
2964:
2960:
2954:
2948:
2945:
2941:
2936:
2932:
2926:
2920:
2917:
2913:
2908:
2901:
2900:
2896:
2890:
2886:
2881:
2876:
2872:
2866:
2860:
2856:
2851:
2846:
2842:
2836:
2830:
2826:
2821:
2816:
2809:
2808:
2806:
2801:
2797:
2793:
2789:
2771:
2770:
2751:
2747:
2743:
2736:
2732:
2728:
2721:
2717:
2711:
2708:
2706:
2696:
2692:
2691:
2684:
2680:
2676:
2669:
2665:
2661:
2654:
2650:
2644:
2641:
2639:
2635:
2631:
2627:
2626:
2619:
2615:
2611:
2604:
2600:
2596:
2589:
2585:
2579:
2576:
2574:
2570:
2566:
2562:
2561:
2554:
2550:
2546:
2539:
2535:
2531:
2524:
2520:
2514:
2511:
2509:
2505:
2501:
2497:
2496:
2481:
2478:
2462:
2459:
2454:
2415:
2411:
2407:
2404:
2381:
2378:
2374:
2370:
2367:
2345:
2342:
2338:
2334:
2331:
2309:
2306:
2302:
2279:
2276:
2272:
2260:
2259:
2258:
2257:
2242:
2237:
2229:
2225:
2219:
2215:
2211:
2204:
2201:
2196:
2192:
2188:
2182:
2178:
2172:
2168:
2162:
2158:
2154:
2149:
2145:
2139:
2135:
2131:
2128:
2121:
2118:
2113:
2109:
2105:
2099:
2095:
2089:
2085:
2079:
2075:
2071:
2066:
2062:
2056:
2052:
2048:
2047:
2042:
2038:
2032:
2028:
2024:
2017:
2014:
2009:
2005:
2001:
1995:
1991:
1985:
1981:
1975:
1971:
1967:
1962:
1958:
1952:
1948:
1944:
1937:
1934:
1929:
1925:
1921:
1915:
1911:
1905:
1901:
1895:
1891:
1887:
1882:
1878:
1872:
1868:
1864:
1861:
1860:
1852:
1849:
1844:
1840:
1837:
1833:
1827:
1823:
1819:
1815:
1811:
1805:
1801:
1797:
1793:
1789:
1783:
1779:
1775:
1774:
1772:
1767:
1764:
1762:
1760:
1755:
1749:
1746:
1744:
1741:
1739:
1736:
1735:
1732:
1729:
1725:
1721:
1717:
1713:
1709:
1705:
1702:
1701:
1698:
1695:
1691:
1687:
1683:
1679:
1675:
1671:
1670:
1668:
1661:
1653:
1649:
1645:
1643:
1640:
1633:
1630:
1625:
1621:
1617:
1611:
1607:
1603:
1600:
1599:
1596:
1593:
1591:
1588:
1586:
1583:
1582:
1574:
1571:
1566:
1562:
1559:
1555:
1549:
1545:
1541:
1539:
1536:
1532:
1528:
1524:
1523:
1521:
1514:
1506:
1502:
1498:
1494:
1490:
1486:
1483:
1481:
1478:
1477:
1472:
1468:
1464:
1460:
1456:
1452:
1450:
1447:
1446:
1443:
1440:
1438:
1435:
1433:
1430:
1429:
1427:
1422:
1420:
1388:
1385:
1380:
1357:
1353:
1330:
1326:
1303:
1299:
1285:
1282:
1264:
1261:
1248:
1247:Standard Model
1245:
1243:
1240:
1203:
1198:
1193:
1189:
1185:
1181:
1157:
1153:
1132:
1120:
1119:
1108:
1100:
1089:
1085:
1078:
1077:
1072:
1068:
1061:
1060:
1055:
1051:
1044:
1043:
1041:
1034:
1026:
1023:
1019:
1012:
1005:
1002:
998:
991:
984:
981:
977:
970:
969:
964:
961:
957:
950:
943:
940:
936:
929:
925:
922:
918:
911:
910:
905:
901:
896:
889:
882:
878:
873:
866:
859:
855:
850:
843:
842:
840:
835:
830:
819:
815:
808:
807:
802:
798:
791:
790:
784:
779:
772:
771:
769:
731:
728:
704:
701:
698:
695:
692:
689:
686:
663:
659:
655:
617:
613:
590:
586:
563:
559:
537:. These three
516:
512:
489:
485:
461:
456:
436:Standard Model
431:
428:
420:Shoichi Sakata
396:quantum states
349:
348:
346:
345:
338:
331:
323:
320:
319:
318:
317:
312:
307:
299:
298:
297:Flavour mixing
294:
293:
292:
291:
290:
289:
275:
266:
262:
251:
243:
234:
233:
232:
228:
215:
177:
176:
172:
171:
170:
169:
160:
151:
148:
135:
126:
114:
113:
109:
108:
107:
106:
96:
87:
78:
69:
66:
50:
49:
42:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4165:
4154:
4151:
4149:
4146:
4144:
4141:
4140:
4138:
4131:
4128:
4124:
4120:
4116:
4111:
4106:
4102:
4098:
4093:
4088:
4084:
4080:
4079:
4058:
4054:
4047:
4045:
4041:
4036:
4032:
4028:
4024:
4020:
4016:
4011:
4006:
4002:
3998:
3997:
3989:
3986:
3981:
3977:
3973:
3969:
3965:
3961:
3957:
3953:
3948:
3943:
3939:
3935:
3928:
3925:
3920:
3916:
3912:
3908:
3904:
3900:
3895:
3890:
3887:(7): 073012.
3886:
3882:
3881:
3873:
3870:
3865:
3861:
3857:
3853:
3848:
3843:
3839:
3835:
3830:
3825:
3821:
3817:
3810:
3807:
3802:
3798:
3797:
3788:
3784:
3783:
3775:
3772:
3766:
3761:
3757:
3753:
3749:
3745:
3744:
3739:
3732:
3729:
3723:
3715:
3711:
3705:
3702:
3695:
3691:
3688:
3686:
3685:Koide formula
3683:
3681:
3678:
3677:
3673:
3652:
3648:
3643:
3632:
3628:
3622:
3618:
3591:
3587:
3583:
3576:
3572:
3568:
3561:
3557:
3551:
3540:
3532:The value of
3531:
3513:
3509:
3505:
3500:
3496:
3476:
3470:
3466:
3462:
3457:
3453:
3433:
3427:
3423:
3419:
3414:
3410:
3401:
3397:
3373:
3368:
3364:
3333:
3328:
3324:
3293:
3288:
3284:
3274:
3273:
3270:
3267:
3266:
3265:
3251:
3241:
3238:
3235:
3224:
3221:
3218:
3207:
3204:
3201:
3188:
3185:
3182:
3171:
3168:
3165:
3154:
3151:
3148:
3135:
3132:
3129:
3118:
3115:
3112:
3101:
3098:
3095:
3085:
3081:
3076:
3060:
3057:
3053:
3032:
3029:
3025:
3004:
3001:
2997:
2974:
2971:
2967:
2946:
2943:
2939:
2918:
2915:
2911:
2888:
2879:
2858:
2849:
2828:
2819:
2804:
2799:
2791:
2778:
2749:
2745:
2741:
2734:
2730:
2726:
2719:
2715:
2709:
2707:
2694:
2682:
2678:
2674:
2667:
2663:
2659:
2652:
2648:
2642:
2640:
2633:
2629:
2617:
2613:
2609:
2602:
2598:
2594:
2587:
2583:
2577:
2575:
2568:
2564:
2552:
2548:
2544:
2537:
2533:
2529:
2522:
2518:
2512:
2510:
2503:
2499:
2487:
2486:
2485:
2479:
2477:
2452:
2443:
2438:
2436:
2413:
2409:
2405:
2402:
2379:
2376:
2372:
2368:
2365:
2343:
2340:
2336:
2332:
2329:
2307:
2304:
2300:
2277:
2274:
2270:
2240:
2235:
2227:
2223:
2217:
2213:
2194:
2190:
2186:
2180:
2176:
2170:
2166:
2160:
2156:
2152:
2147:
2143:
2137:
2133:
2129:
2111:
2107:
2103:
2097:
2093:
2087:
2083:
2077:
2073:
2069:
2064:
2060:
2054:
2050:
2040:
2036:
2030:
2026:
2007:
2003:
1999:
1993:
1989:
1983:
1979:
1973:
1969:
1965:
1960:
1956:
1950:
1946:
1927:
1923:
1919:
1913:
1909:
1903:
1899:
1893:
1889:
1885:
1880:
1876:
1870:
1866:
1862:
1842:
1838:
1835:
1831:
1825:
1821:
1813:
1809:
1803:
1799:
1791:
1787:
1781:
1777:
1770:
1765:
1763:
1753:
1747:
1742:
1737:
1730:
1723:
1719:
1711:
1707:
1703:
1696:
1689:
1685:
1677:
1673:
1666:
1659:
1651:
1647:
1641:
1623:
1619:
1615:
1609:
1605:
1601:
1594:
1589:
1584:
1564:
1560:
1557:
1553:
1547:
1543:
1537:
1530:
1526:
1519:
1512:
1504:
1500:
1492:
1488:
1484:
1479:
1470:
1466:
1458:
1454:
1448:
1441:
1436:
1431:
1425:
1411:
1410:
1409:
1408:
1407:
1405:
1378:
1355:
1351:
1328:
1324:
1301:
1297:
1283:
1281:
1277:
1275:
1271:
1270:Majorana mass
1262:
1260:
1256:
1254:
1246:
1241:
1239:
1237:
1233:
1228:
1226:
1222:
1221:antineutrinos
1217:
1201:
1196:
1191:
1187:
1183:
1179:
1155:
1151:
1130:
1106:
1098:
1087:
1083:
1070:
1066:
1053:
1049:
1039:
1032:
1024:
1021:
1017:
1003:
1000:
996:
982:
979:
975:
962:
959:
955:
941:
938:
934:
923:
920:
916:
903:
894:
880:
871:
857:
848:
838:
833:
828:
817:
813:
800:
796:
777:
767:
758:
757:
756:
729:
726:
702:
699:
696:
693:
690:
687:
684:
661:
657:
653:
644:
643:superposition
639:
637:
633:
615:
611:
588:
584:
561:
557:
548:
544:
540:
536:
532:
514:
510:
487:
483:
454:
445:
441:
437:
429:
427:
425:
421:
417:
413:
409:
405:
401:
397:
393:
392:mixing matrix
390:
386:
385:mixing matrix
384:
379:
378:mixing matrix
377:
372:
368:
364:
360:
356:
344:
339:
337:
332:
330:
325:
324:
322:
321:
316:
313:
311:
308:
306:
303:
302:
301:
300:
295:
287:
286:
282:
274:
270:
267:
261:
257:
250:
247:
246:
242:
238:
235:
227:
223:
219:
216:
213:
209:
205:
201:
197:
193:
190:
189:
188:
184:
181:
180:
179:
178:
173:
168:
164:
161:
159:
155:
152:
147:
143:
139:
136:
134:
130:
129:Lepton number
127:
125:
121:
120:Baryon number
118:
117:
116:
115:
110:
104:
100:
97:
95:
91:
88:
86:
82:
79:
77:
73:
70:
65:
61:
57:
54:
53:
52:
51:
48:
43:
40:
35:
31:
19:
4082:
4076:
4073:
4061:. Retrieved
4056:
4000:
3994:
3988:
3937:
3933:
3927:
3884:
3878:
3872:
3819:
3815:
3809:
3800:
3794:
3786:
3780:
3774:
3747:
3741:
3731:
3714:seesaw model
3709:
3704:
2779:
2772:
2483:
2439:
2261:
1287:
1278:
1272:rather than
1266:
1263:Other models
1257:
1250:
1229:
1225:CPT symmetry
1218:
1121:
640:
433:
381:
374:
370:
366:
362:
358:
352:
309:
284:
280:
272:
268:
259:
255:
248:
240:
225:
221:
217:
211:
207:
203:
199:
195:
191:
186:
175:Combinations
166:
157:
145:
141:
138:Weak isospin
132:
123:
102:
93:
84:
75:
63:
59:
4085:(12): 123.
3816:Dark Matter
1402:related to
1242:Assumptions
632:Hamiltonian
539:eigenstates
440:generations
363:PMNS matrix
310:PMNS matrix
183:Hypercharge
81:Strangeness
4137:Categories
3750:(5): 870.
3724:References
1274:Dirac mass
1236:CKM matrix
547:eigenbasis
371:MNS matrix
305:CKM matrix
210:′ +
99:Bottomness
4143:Neutrinos
4127:118566415
4105:CiteSeerX
4092:1209.3023
4057:NuFIT.org
3947:1407.7259
3919:119208413
3894:1308.6218
3864:119182490
3842:CiteSeerX
3829:1402.3028
3653:∘
3644:≤
3633:δ
3629:≤
3623:∘
3592:∘
3577:∘
3569:−
3562:∘
3541:δ
3514:∘
3497:θ
3471:∘
3454:θ
3428:∘
3420:≈
3411:θ
3365:θ
3325:θ
3285:θ
3239:∼
3222:∼
3205:∼
3186:∼
3169:∼
3152:∼
3133:∼
3116:∼
3099:∼
3058:τ
3030:τ
3002:τ
2972:μ
2944:μ
2916:μ
2750:∘
2735:∘
2727:−
2720:∘
2695:δ
2683:∘
2668:∘
2660:−
2653:∘
2630:θ
2618:∘
2603:∘
2595:−
2588:∘
2565:θ
2553:∘
2538:∘
2530:−
2523:∘
2500:θ
2453:δ
2410:ν
2403:ν
2373:θ
2369:
2337:θ
2333:
2195:δ
2153:−
2130:−
2112:δ
2070:−
2008:δ
1966:−
1928:δ
1886:−
1863:−
1843:δ
1836:−
1704:−
1624:δ
1602:−
1565:δ
1558:−
1485:−
1379:δ
1352:θ
1325:θ
1298:θ
1188:α
1131:α
1084:ν
1067:ν
1050:ν
1022:τ
1001:τ
980:τ
960:μ
939:μ
921:μ
818:τ
814:ν
801:μ
797:ν
778:ν
727:α
658:α
612:ν
585:ν
558:ν
515:τ
511:ν
488:μ
484:ν
455:ν
412:Ziro Maki
400:neutrinos
4063:29 March
3980:10500157
3972:25325631
3674:See also
383:neutrino
163:X-charge
45:Flavour
4148:Leptons
4097:Bibcode
4035:2094005
4015:Bibcode
3952:Bibcode
3899:Bibcode
3834:Bibcode
3752:Bibcode
1253:unitary
444:flavors
389:unitary
105:′
90:Topness
56:Isospin
34:Flavour
4125:
4107:
4033:
3978:
3970:
3917:
3862:
3844:
3803:: 172.
3789:: 247.
3402:(i.e.
3392:0.011°
3388:0.201°
3308:13.04°
3245:
3231:
3228:
3214:
3211:
3199:
3192:
3178:
3175:
3161:
3158:
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3122:
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3105:
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1343:, and
1104:
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988:
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810:
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774:
724:
636:quarks
603:, and
531:lepton
502:, and
418:, and
376:lepton
357:, the
4123:S2CID
4087:arXiv
4031:S2CID
4005:arXiv
3976:S2CID
3942:arXiv
3915:S2CID
3889:arXiv
3860:S2CID
3824:arXiv
3696:Notes
3352:0.06°
3348:2.38°
3312:0.05°
3242:0.762
3236:0.610
3225:0.702
3219:0.473
3208:0.525
3202:0.262
3189:0.779
3183:0.630
3172:0.693
3166:0.460
3155:0.505
3149:0.233
3136:0.155
3130:0.142
3119:0.578
3113:0.514
3102:0.845
3096:0.803
2519:33.41
387:is a
380:, or
278:= 5 (
254:= 2 (
220:= 2 (
72:Charm
4083:2012
4065:2023
3968:PMID
3820:1604
3424:35.3
2679:0.11
2664:0.12
2649:8.54
2584:49.1
2549:0.75
2534:0.72
2358:and
2292:and
751:", "
747:", "
442:or "
434:The
4115:doi
4023:doi
3960:doi
3938:113
3907:doi
3852:doi
3760:doi
3710:not
3649:246
3619:169
3558:197
2716:197
2614:1.0
2599:1.3
2366:cos
2330:sin
1238:).
398:of
373:),
365:),
353:In
271:+ 2
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144:or
62:or
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2228:13
2218:23
2181:13
2171:23
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2148:23
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2078:12
2065:23
2055:12
2041:13
2031:23
1994:13
1984:23
1974:12
1961:23
1951:12
1914:13
1904:23
1894:12
1881:23
1871:12
1826:13
1814:13
1804:12
1792:13
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1712:12
1690:12
1678:12
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1302:12
1276:.
1227:.
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475:,
426:.
414:,
283:−
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3584:+
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3506:=
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3463:=
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3390:±
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3294:=
3252:]
3086:[
3082:=
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3067:|
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2800:=
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2431:.
2414:c
2406:=
2380:j
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2214:c
2203:P
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2094:s
2084:c
2074:c
2061:s
2051:s
2037:c
2027:s
2016:P
2013:C
2004:i
2000:e
1990:s
1980:s
1970:s
1957:c
1947:c
1936:P
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1910:s
1900:s
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1771:[
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