2524:
2082:
2519:{\displaystyle I=\int _{0}^{\infty }{\frac {dp}{(2\pi )^{4-\varepsilon }}}{\frac {2\pi ^{(4-\varepsilon )/2}}{\Gamma \left({\frac {4-\varepsilon }{2}}\right)}}{\frac {p^{3-\varepsilon }}{\left(p^{2}+m^{2}\right)^{2}}}={\frac {2^{\varepsilon -4}\pi ^{{\frac {\varepsilon }{2}}-1}}{\sin \left({\frac {\pi \varepsilon }{2}}\right)\Gamma \left(1-{\frac {\varepsilon }{2}}\right)}}m^{-\varepsilon }={\frac {1}{8\pi ^{2}\varepsilon }}-{\frac {1}{16\pi ^{2}}}\left(\ln {\frac {m^{2}}{4\pi }}+\gamma \right)+{\mathcal {O}}(\varepsilon ).}
710:
1308:
461:
1061:
1973:
1098:
1781:
1610:
249:. A further leap is to take the interpolation through fractional dimensions seriously. This has led some authors to suggest that dimensional regularization can be used to study the physics of crystals that macroscopically appear to be
705:{\displaystyle \int _{-\infty }^{\infty }{\frac {dy}{\sqrt {x^{2}+y^{2}}}}=\int _{-\infty }^{\infty }{\frac {dt}{\sqrt {(x/x_{0})^{2}+t^{2}}}}=\int _{0}^{\infty }{\frac {\mathrm {vol} (S^{1})dr}{\sqrt {(x/x_{0})^{2}+r^{2}}}}}
398:
911:
1433:
1813:
2073:
882:
2550:
454:
1670:
1483:
82:
2576:
1358:
1645:
766:
1093:
458:
Since the charged line has 1-dimensional "spherical symmetry" (which in 1-dimension is just mirror symmetry), we can rewrite the integral to exploit the spherical symmetry:
1665:
108:
1303:{\displaystyle \int _{0}^{\infty }{\frac {r^{d-1}dr}{\sqrt {(x/x_{0})^{2}+r^{2}}}}\sim \int _{c}^{\infty }r^{d-2}dr={\frac {1}{d-1}}c^{d-1}=\epsilon ^{-1}c^{-\epsilon },}
313:
902:
1808:
1499:
793:
737:
1999:
816:
308:
288:
260:
and dimensional regularization are equivalent since they use the same principle of using analytic continuation in order for a series or integral to converge.
101:
134:
2844:
94:
230:
showed that dimensional regularization is mathematically well defined, at least in the case of massive
Euclidean fields, by using the
1367:
2915:
2820:
2004:
1493:
Suppose one wishes to dimensionally regularize a loop integral which is logarithmically divergent in four dimensions, like
821:
1056:{\displaystyle {\frac {2\pi ^{d/2}}{\Gamma (d/2)}}\int _{0}^{\infty }{\frac {r^{d-1}dr}{\sqrt {(x/x_{0})^{2}+r^{2}}}}}
2740:
62:
245:
is taken to approach another integer value where the theory appears to be strongly coupled as in the case of the
38:
1968:{\displaystyle \int d^{d}p\,f(p^{2})={\frac {2\pi ^{d/2}}{\Gamma (d/2)}}\int _{0}^{\infty }dp\,p^{d-1}f(p^{2}).}
2529:
403:
2754:"Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&M"
2991:
231:
2996:
257:
70:
2555:
1313:
74:
42:
1618:
1438:
742:
1066:
246:
146:
53:
1650:
78:
211:
142:
66:
1776:{\displaystyle I=\int {\frac {d^{d}p}{(2\pi )^{d}}}{\frac {1}{\left(p^{2}+m^{2}\right)^{2}}}.}
1605:{\displaystyle I=\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {1}{\left(p^{2}+m^{2}\right)^{2}}}.}
130:
887:
34:
2753:
2945:
2925:
2875:
2666:
2617:
1786:
771:
715:
157:
2908:
Quantum fields and strings: a course for mathematicians, Vol. 1,(Princeton, NJ, 1996/1997)
8:
2710:
1978:
122:
2949:
2879:
2670:
2621:
2891:
2863:
2838:
2791:
2765:
1361:
801:
293:
273:
138:
2971:
2957:
2911:
2895:
2826:
2816:
2783:
2736:
2692:
2678:
2635:
250:
2933:
2864:"Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter."
2795:
2961:
2953:
2883:
2775:
2682:
2674:
2625:
168:
2921:
393:{\displaystyle V(x)=A\int _{-\infty }^{\infty }{\frac {dy}{\sqrt {x^{2}+y^{2}}}}}
223:
203:
153:
29:
2654:
905:
712:
where we first removed the dependence on length by dividing with a unit-length
2830:
2711:"Accurate critical exponents for Ising-like systems in non-integer dimensions"
2985:
2975:
2787:
2696:
2639:
2810:
2079:
the value of the integral in this way by analytic continuation. This gives
237:
Although the method is most well understood when poles are subtracted and
1615:
First, write the integral in a general non-integer number of dimensions
218:. In general, there will be a pole at the physical value (usually 4) of
2887:
2779:
2630:
2605:
2966:
2687:
2903:
241:
is once again replaced by 4, it has also led to some successes when
2001:, this formula reduces to familiar integrals over thin shells like
214:
from this region to a meromorphic function defined for all complex
164:, the analytic continuation of the number of spacetime dimensions.
149:
2770:
2731:
A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini,
263:
795:, followed by an integral over all radii of the 1-sphere.
137:
as well as – independently and more comprehensively – by
2910:, Providence, R.I.: Amer. Math. Soc., pp. 597–607,
1428:{\displaystyle V(x)\sim (x_{0}/x)^{\epsilon }/\epsilon }
2068:{\textstyle \int _{0}^{\infty }dp\,4\pi p^{2}f(p^{2})}
2007:
270:
Consider an infinite charged line with charge density
2708:
2558:
2532:
2085:
1981:
1816:
1789:
1673:
1653:
1621:
1502:
1441:
1370:
1316:
1101:
1069:
914:
890:
824:
804:
774:
745:
718:
464:
406:
316:
296:
290:, and we calculate the potential of a point distance
276:
2934:"Regularization and renormalization of gauge fields"
2655:"Regularization and renormalization of gauge fields"
171:
as an integral depending on the spacetime dimension
156:; in other words, assigning values to them that are
877:{\displaystyle {\frac {2\pi ^{d/2}}{\Gamma (d/2)}}}
2604:Bietenholz, Wolfgang; Prado, Lilian (2014-02-01).
2570:
2544:
2518:
2067:
1993:
1967:
1802:
1775:
1659:
1639:
1604:
1477:
1427:
1352:
1302:
1095:, the integral is dominated by its tail, that is,
1087:
1055:
896:
876:
810:
787:
760:
731:
704:
448:
392:
302:
282:
2752:Olness, Fredrick; Scalise, Randall (March 2011).
2983:
2861:
2606:"Revolutionary physics in reactionary Argentina"
2603:
2862:Bollini, Carlos; Giambiagi, Juan Jose (1972),
264:Example: potential of an infinite charged line
2751:
206:, the integral often converges for −Re(
102:
2931:
2652:
2597:
2552:, but is finite for arbitrary small values
2843:: CS1 maint: location missing publisher (
2709:Le Guillou, J.C.; Zinn-Justin, J. (1987).
310:away from the line. The integral diverges:
109:
95:
2965:
2769:
2686:
2629:
2545:{\displaystyle \varepsilon \rightarrow 0}
2526:Note that the integral again diverges as
2029:
1926:
1833:
748:
449:{\displaystyle A=s/(4\pi \epsilon _{0}).}
234:to carry out the analytic continuation.
16:Method in evaluating divergent integrals
2901:
2812:An introduction to quantum field theory
227:
2984:
2808:
798:Now we generalize this into dimension
2735:, World Scientific Publishing, 2003,
2904:"Note on dimensional regularization"
167:Dimensional regularization writes a
2815:. Daniel V. Schroeder. Boca Raton.
768:into an integral over the 1-sphere
739:, then converted the integral over
13:
2932:Hooft, G. 't; Veltman, M. (1972),
2855:
2653:Hooft, G. 't; Veltman, M. (1972),
2571:{\displaystyle \varepsilon \neq 0}
2499:
2352:
2184:
2102:
2018:
1915:
1882:
1353:{\displaystyle c=\Theta (x/x_{0})}
1323:
1204:
1112:
974:
941:
891:
851:
627:
624:
621:
612:
536:
531:
478:
473:
348:
343:
22:Renormalization and regularization
14:
3008:
2733:Analytic Aspects of Quantum Field
2075:. For non-integer dimensions, we
1783:If the integrand only depends on
210:) sufficiently large, and can be
1667:will later be taken to be small,
1640:{\displaystyle d=4-\varepsilon }
1478:{\displaystyle V'(x)\sim x^{-1}}
761:{\displaystyle \mathbb {R} ^{1}}
222:, which needs to be canceled by
2809:Peskin, Michael Edward (2019).
1435:, and so the electric field is
226:to obtain physical quantities.
2802:
2745:
2725:
2702:
2646:
2588:
2536:
2510:
2504:
2169:
2157:
2128:
2118:
2062:
2049:
1959:
1946:
1899:
1885:
1850:
1837:
1711:
1701:
1540:
1530:
1456:
1450:
1408:
1386:
1380:
1374:
1347:
1326:
1166:
1144:
1028:
1006:
958:
944:
868:
854:
818:. The volume of a d-sphere is
677:
655:
644:
631:
574:
552:
440:
421:
326:
320:
83:Point-splitting regularization
1:
2581:
1088:{\displaystyle d=1-\epsilon }
2958:10.1016/0550-3213(72)90279-9
2679:10.1016/0550-3213(72)90279-9
1975:For integer dimensions like
1660:{\displaystyle \varepsilon }
258:Zeta function regularization
71:Zeta function regularization
63:Pauli–Villars regularization
7:
2758:American Journal of Physics
908:. Now the integral becomes
202:, ... appearing in it. In
175:and the squared distances (
10:
3013:
1810:, we can apply the formula
1488:
193:) of the spacetime points
129:is a method introduced by
127:dimensional regularization
75:Causal perturbation theory
59:Dimensional regularization
43:Minimal subtraction scheme
247:Wilson–Fisher fixed point
232:Bernstein–Sato polynomial
256:It has been argued that
2902:Etingof, Pavel (1999),
897:{\displaystyle \Gamma }
160:of a complex parameter
79:Hadamard regularization
2572:
2546:
2520:
2069:
1995:
1969:
1804:
1777:
1661:
1641:
1606:
1479:
1429:
1354:
1304:
1089:
1057:
898:
878:
812:
789:
762:
733:
706:
450:
394:
304:
284:
212:analytically continued
67:Lattice regularization
2573:
2547:
2521:
2070:
1996:
1970:
1805:
1803:{\displaystyle p^{2}}
1778:
1662:
1642:
1607:
1480:
1430:
1355:
1305:
1090:
1058:
899:
879:
813:
790:
788:{\displaystyle S^{1}}
763:
734:
732:{\displaystyle x_{0}}
707:
451:
395:
305:
285:
158:meromorphic functions
152:in the evaluation of
35:Renormalization group
2992:Quantum field theory
2594:Bollini 1972, p. 20.
2556:
2530:
2083:
2005:
1979:
1814:
1787:
1671:
1651:
1619:
1500:
1439:
1368:
1314:
1099:
1067:
912:
888:
822:
802:
772:
743:
716:
462:
404:
314:
294:
274:
2997:Summability methods
2950:1972NuPhB..44..189T
2880:1972NCimB..12...20B
2715:Journal de Physique
2671:1972NuPhB..44..189T
2622:2014PhT....67b..38B
2106:
2022:
1994:{\displaystyle d=3}
1919:
1208:
1116:
978:
616:
540:
482:
352:
123:theoretical physics
2888:10.1007/BF02895558
2868:Il Nuovo Cimento B
2568:
2542:
2516:
2092:
2065:
2008:
1991:
1965:
1905:
1800:
1773:
1657:
1637:
1602:
1475:
1425:
1362:big theta notation
1350:
1300:
1194:
1102:
1085:
1053:
964:
894:
874:
808:
785:
758:
729:
702:
602:
523:
465:
446:
390:
335:
300:
280:
2938:Nuclear Physics B
2917:978-0-8218-2012-4
2822:978-0-201-50397-5
2780:10.1119/1.3535586
2659:Nuclear Physics B
2631:10.1063/PT.3.2277
2481:
2448:
2423:
2382:
2374:
2346:
2311:
2274:
2214:
2207:
2144:
1903:
1768:
1721:
1597:
1550:
1250:
1189:
1188:
1051:
1050:
962:
872:
811:{\displaystyle d}
700:
699:
597:
596:
518:
517:
388:
387:
303:{\displaystyle x}
283:{\displaystyle s}
119:
118:
3004:
2978:
2969:
2928:
2898:
2849:
2848:
2842:
2834:
2806:
2800:
2799:
2773:
2749:
2743:
2729:
2723:
2722:
2706:
2700:
2699:
2690:
2650:
2644:
2643:
2633:
2601:
2595:
2592:
2577:
2575:
2574:
2569:
2551:
2549:
2548:
2543:
2525:
2523:
2522:
2517:
2503:
2502:
2493:
2489:
2482:
2480:
2472:
2471:
2462:
2449:
2447:
2446:
2445:
2429:
2424:
2422:
2418:
2417:
2401:
2396:
2395:
2383:
2381:
2380:
2376:
2375:
2367:
2351:
2347:
2342:
2334:
2321:
2320:
2319:
2312:
2304:
2297:
2296:
2280:
2275:
2273:
2272:
2267:
2263:
2262:
2261:
2249:
2248:
2233:
2232:
2217:
2215:
2213:
2212:
2208:
2203:
2192:
2182:
2181:
2180:
2176:
2147:
2145:
2143:
2142:
2141:
2116:
2108:
2105:
2100:
2074:
2072:
2071:
2066:
2061:
2060:
2045:
2044:
2021:
2016:
2000:
1998:
1997:
1992:
1974:
1972:
1971:
1966:
1958:
1957:
1942:
1941:
1918:
1913:
1904:
1902:
1895:
1880:
1879:
1878:
1874:
1857:
1849:
1848:
1829:
1828:
1809:
1807:
1806:
1801:
1799:
1798:
1782:
1780:
1779:
1774:
1769:
1767:
1766:
1761:
1757:
1756:
1755:
1743:
1742:
1724:
1722:
1720:
1719:
1718:
1699:
1695:
1694:
1684:
1666:
1664:
1663:
1658:
1646:
1644:
1643:
1638:
1611:
1609:
1608:
1603:
1598:
1596:
1595:
1590:
1586:
1585:
1584:
1572:
1571:
1553:
1551:
1549:
1548:
1547:
1528:
1524:
1523:
1513:
1485:, as it should.
1484:
1482:
1481:
1476:
1474:
1473:
1449:
1434:
1432:
1431:
1426:
1421:
1416:
1415:
1403:
1398:
1397:
1359:
1357:
1356:
1351:
1346:
1345:
1336:
1309:
1307:
1306:
1301:
1296:
1295:
1283:
1282:
1267:
1266:
1251:
1249:
1235:
1224:
1223:
1207:
1202:
1190:
1187:
1186:
1174:
1173:
1164:
1163:
1154:
1143:
1142:
1135:
1134:
1118:
1115:
1110:
1094:
1092:
1091:
1086:
1062:
1060:
1059:
1054:
1052:
1049:
1048:
1036:
1035:
1026:
1025:
1016:
1005:
1004:
997:
996:
980:
977:
972:
963:
961:
954:
939:
938:
937:
933:
916:
903:
901:
900:
895:
883:
881:
880:
875:
873:
871:
864:
849:
848:
847:
843:
826:
817:
815:
814:
809:
794:
792:
791:
786:
784:
783:
767:
765:
764:
759:
757:
756:
751:
738:
736:
735:
730:
728:
727:
711:
709:
708:
703:
701:
698:
697:
685:
684:
675:
674:
665:
654:
653:
643:
642:
630:
618:
615:
610:
598:
595:
594:
582:
581:
572:
571:
562:
551:
550:
542:
539:
534:
519:
516:
515:
503:
502:
493:
492:
484:
481:
476:
455:
453:
452:
447:
439:
438:
420:
399:
397:
396:
391:
389:
386:
385:
373:
372:
363:
362:
354:
351:
346:
309:
307:
306:
301:
289:
287:
286:
281:
169:Feynman integral
154:Feynman diagrams
111:
104:
97:
19:
18:
3012:
3011:
3007:
3006:
3005:
3003:
3002:
3001:
2982:
2981:
2918:
2858:
2856:Further reading
2853:
2852:
2836:
2835:
2823:
2807:
2803:
2750:
2746:
2730:
2726:
2707:
2703:
2651:
2647:
2602:
2598:
2593:
2589:
2584:
2557:
2554:
2553:
2531:
2528:
2527:
2498:
2497:
2473:
2467:
2463:
2461:
2454:
2450:
2441:
2437:
2433:
2428:
2413:
2409:
2405:
2400:
2388:
2384:
2366:
2359:
2355:
2335:
2333:
2329:
2322:
2303:
2302:
2298:
2286:
2282:
2281:
2279:
2268:
2257:
2253:
2244:
2240:
2239:
2235:
2234:
2222:
2218:
2216:
2193:
2191:
2187:
2183:
2172:
2156:
2152:
2148:
2146:
2131:
2127:
2117:
2109:
2107:
2101:
2096:
2084:
2081:
2080:
2056:
2052:
2040:
2036:
2017:
2012:
2006:
2003:
2002:
1980:
1977:
1976:
1953:
1949:
1931:
1927:
1914:
1909:
1891:
1881:
1870:
1866:
1862:
1858:
1856:
1844:
1840:
1824:
1820:
1815:
1812:
1811:
1794:
1790:
1788:
1785:
1784:
1762:
1751:
1747:
1738:
1734:
1733:
1729:
1728:
1723:
1714:
1710:
1700:
1690:
1686:
1685:
1683:
1672:
1669:
1668:
1652:
1649:
1648:
1620:
1617:
1616:
1591:
1580:
1576:
1567:
1563:
1562:
1558:
1557:
1552:
1543:
1539:
1529:
1519:
1515:
1514:
1512:
1501:
1498:
1497:
1491:
1466:
1462:
1442:
1440:
1437:
1436:
1417:
1411:
1407:
1399:
1393:
1389:
1369:
1366:
1365:
1341:
1337:
1332:
1315:
1312:
1311:
1288:
1284:
1275:
1271:
1256:
1252:
1239:
1234:
1213:
1209:
1203:
1198:
1182:
1178:
1169:
1165:
1159:
1155:
1150:
1124:
1120:
1119:
1117:
1111:
1106:
1100:
1097:
1096:
1068:
1065:
1064:
1044:
1040:
1031:
1027:
1021:
1017:
1012:
986:
982:
981:
979:
973:
968:
950:
940:
929:
925:
921:
917:
915:
913:
910:
909:
889:
886:
885:
860:
850:
839:
835:
831:
827:
825:
823:
820:
819:
803:
800:
799:
779:
775:
773:
770:
769:
752:
747:
746:
744:
741:
740:
723:
719:
717:
714:
713:
693:
689:
680:
676:
670:
666:
661:
638:
634:
620:
619:
617:
611:
606:
590:
586:
577:
573:
567:
563:
558:
543:
541:
535:
527:
511:
507:
498:
494:
485:
483:
477:
469:
463:
460:
459:
434:
430:
416:
405:
402:
401:
381:
377:
368:
364:
355:
353:
347:
339:
315:
312:
311:
295:
292:
291:
275:
272:
271:
266:
224:renormalization
204:Euclidean space
201:
192:
183:
115:
86:
85:
81:
77:
73:
69:
65:
61:
56:
46:
45:
41:
39:On-shell scheme
37:
32:
30:Renormalization
17:
12:
11:
5:
3010:
3000:
2999:
2994:
2980:
2979:
2944:(1): 189–213,
2929:
2916:
2899:
2857:
2854:
2851:
2850:
2821:
2801:
2764:(3): 306–312.
2744:
2724:
2701:
2665:(1): 189–213,
2645:
2596:
2586:
2585:
2583:
2580:
2567:
2564:
2561:
2541:
2538:
2535:
2515:
2512:
2509:
2506:
2501:
2496:
2492:
2488:
2485:
2479:
2476:
2470:
2466:
2460:
2457:
2453:
2444:
2440:
2436:
2432:
2427:
2421:
2416:
2412:
2408:
2404:
2399:
2394:
2391:
2387:
2379:
2373:
2370:
2365:
2362:
2358:
2354:
2350:
2345:
2341:
2338:
2332:
2328:
2325:
2318:
2315:
2310:
2307:
2301:
2295:
2292:
2289:
2285:
2278:
2271:
2266:
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2252:
2247:
2243:
2238:
2231:
2228:
2225:
2221:
2211:
2206:
2202:
2199:
2196:
2190:
2186:
2179:
2175:
2171:
2168:
2165:
2162:
2159:
2155:
2151:
2140:
2137:
2134:
2130:
2126:
2123:
2120:
2115:
2112:
2104:
2099:
2095:
2091:
2088:
2064:
2059:
2055:
2051:
2048:
2043:
2039:
2035:
2032:
2028:
2025:
2020:
2015:
2011:
1990:
1987:
1984:
1964:
1961:
1956:
1952:
1948:
1945:
1940:
1937:
1934:
1930:
1925:
1922:
1917:
1912:
1908:
1901:
1898:
1894:
1890:
1887:
1884:
1877:
1873:
1869:
1865:
1861:
1855:
1852:
1847:
1843:
1839:
1836:
1832:
1827:
1823:
1819:
1797:
1793:
1772:
1765:
1760:
1754:
1750:
1746:
1741:
1737:
1732:
1727:
1717:
1713:
1709:
1706:
1703:
1698:
1693:
1689:
1682:
1679:
1676:
1656:
1636:
1633:
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1627:
1624:
1613:
1612:
1601:
1594:
1589:
1583:
1579:
1575:
1570:
1566:
1561:
1556:
1546:
1542:
1538:
1535:
1532:
1527:
1522:
1518:
1511:
1508:
1505:
1490:
1487:
1472:
1469:
1465:
1461:
1458:
1455:
1452:
1448:
1445:
1424:
1420:
1414:
1410:
1406:
1402:
1396:
1392:
1388:
1385:
1382:
1379:
1376:
1373:
1349:
1344:
1340:
1335:
1331:
1328:
1325:
1322:
1319:
1299:
1294:
1291:
1287:
1281:
1278:
1274:
1270:
1265:
1262:
1259:
1255:
1248:
1245:
1242:
1238:
1233:
1230:
1227:
1222:
1219:
1216:
1212:
1206:
1201:
1197:
1193:
1185:
1181:
1177:
1172:
1168:
1162:
1158:
1153:
1149:
1146:
1141:
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1130:
1127:
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1114:
1109:
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1081:
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1047:
1043:
1039:
1034:
1030:
1024:
1020:
1015:
1011:
1008:
1003:
1000:
995:
992:
989:
985:
976:
971:
967:
960:
957:
953:
949:
946:
943:
936:
932:
928:
924:
920:
906:gamma function
893:
870:
867:
863:
859:
856:
853:
846:
842:
838:
834:
830:
807:
782:
778:
755:
750:
726:
722:
696:
692:
688:
683:
679:
673:
669:
664:
660:
657:
652:
649:
646:
641:
637:
633:
629:
626:
623:
614:
609:
605:
601:
593:
589:
585:
580:
576:
570:
566:
561:
557:
554:
549:
546:
538:
533:
530:
526:
522:
514:
510:
506:
501:
497:
491:
488:
480:
475:
472:
468:
445:
442:
437:
433:
429:
426:
423:
419:
415:
412:
409:
384:
380:
376:
371:
367:
361:
358:
350:
345:
342:
338:
334:
331:
328:
325:
322:
319:
299:
279:
265:
262:
228:Etingof (1999)
197:
188:
179:
117:
116:
114:
113:
106:
99:
91:
88:
87:
57:
54:Regularization
52:
51:
48:
47:
33:
28:
27:
24:
23:
15:
9:
6:
4:
3:
2:
3009:
2998:
2995:
2993:
2990:
2989:
2987:
2977:
2973:
2968:
2963:
2959:
2955:
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2947:
2943:
2939:
2935:
2930:
2927:
2923:
2919:
2913:
2909:
2905:
2900:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2865:
2860:
2859:
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2840:
2832:
2828:
2824:
2818:
2814:
2813:
2805:
2797:
2793:
2789:
2785:
2781:
2777:
2772:
2767:
2763:
2759:
2755:
2748:
2742:
2741:981-238-364-6
2738:
2734:
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2716:
2712:
2705:
2698:
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2689:
2684:
2680:
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2672:
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2664:
2660:
2656:
2649:
2641:
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2632:
2627:
2623:
2619:
2615:
2611:
2610:Physics Today
2607:
2600:
2591:
2587:
2579:
2565:
2562:
2559:
2539:
2533:
2513:
2507:
2494:
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2290:
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2254:
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2236:
2229:
2226:
2223:
2219:
2209:
2204:
2200:
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2194:
2188:
2177:
2173:
2166:
2163:
2160:
2153:
2149:
2138:
2135:
2132:
2124:
2121:
2113:
2110:
2097:
2093:
2089:
2086:
2078:
2057:
2053:
2046:
2041:
2037:
2033:
2030:
2026:
2023:
2013:
2009:
1988:
1985:
1982:
1962:
1954:
1950:
1943:
1938:
1935:
1932:
1928:
1923:
1920:
1910:
1906:
1896:
1892:
1888:
1875:
1871:
1867:
1863:
1859:
1853:
1845:
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1758:
1752:
1748:
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1536:
1533:
1525:
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1459:
1453:
1446:
1443:
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1412:
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1390:
1383:
1377:
1371:
1363:
1342:
1338:
1333:
1329:
1320:
1317:
1297:
1292:
1289:
1285:
1279:
1276:
1272:
1268:
1263:
1260:
1257:
1253:
1246:
1243:
1240:
1236:
1231:
1228:
1225:
1220:
1217:
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1191:
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1156:
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1131:
1128:
1125:
1121:
1107:
1103:
1082:
1079:
1076:
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1070:
1045:
1041:
1037:
1032:
1022:
1018:
1013:
1009:
1001:
998:
993:
990:
987:
983:
969:
965:
955:
951:
947:
934:
930:
926:
922:
918:
907:
865:
861:
857:
844:
840:
836:
832:
828:
805:
796:
780:
776:
753:
724:
720:
694:
690:
686:
681:
671:
667:
662:
658:
650:
647:
639:
635:
607:
603:
599:
591:
587:
583:
578:
568:
564:
559:
555:
547:
544:
528:
524:
520:
512:
508:
504:
499:
495:
489:
486:
470:
466:
456:
443:
435:
431:
427:
424:
417:
413:
410:
407:
382:
378:
374:
369:
365:
359:
356:
340:
336:
332:
329:
323:
317:
297:
277:
268:
261:
259:
254:
252:
248:
244:
240:
235:
233:
229:
225:
221:
217:
213:
209:
205:
200:
196:
191:
187:
182:
178:
174:
170:
165:
163:
159:
155:
151:
148:
144:
140:
136:
132:
128:
124:
112:
107:
105:
100:
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93:
92:
90:
89:
84:
80:
76:
72:
68:
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50:
49:
44:
40:
36:
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2937:
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2871:
2867:
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2804:
2761:
2757:
2747:
2732:
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2704:
2662:
2658:
2648:
2616:(2): 38–43.
2613:
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2076:
1614:
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457:
269:
267:
255:
242:
238:
236:
219:
215:
207:
198:
194:
189:
185:
180:
176:
172:
166:
161:
147:regularizing
126:
120:
58:
2986:Categories
2831:1101381398
2582:References
2976:0550-3213
2967:1874/4845
2896:123505054
2839:cite book
2788:0002-9505
2771:0812.3578
2697:0550-3213
2688:1874/4845
2640:0031-9228
2563:≠
2560:ε
2537:→
2534:ε
2508:ε
2487:γ
2478:π
2459:
2439:π
2426:−
2420:ε
2411:π
2393:ε
2390:−
2369:ε
2364:−
2353:Γ
2340:ε
2337:π
2327:
2314:−
2306:ε
2300:π
2291:−
2288:ε
2230:ε
2227:−
2201:ε
2198:−
2185:Γ
2167:ε
2164:−
2154:π
2139:ε
2136:−
2125:π
2103:∞
2094:∫
2034:π
2019:∞
2010:∫
1936:−
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1907:∫
1883:Γ
1864:π
1818:∫
1708:π
1681:∫
1655:ε
1635:ε
1632:−
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1510:∫
1468:−
1460:∼
1423:ϵ
1413:ϵ
1384:∼
1324:Θ
1293:ϵ
1290:−
1277:−
1273:ϵ
1261:−
1244:−
1218:−
1205:∞
1196:∫
1192:∼
1129:−
1113:∞
1104:∫
1083:ϵ
1080:−
991:−
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966:∫
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892:Γ
852:Γ
833:π
613:∞
604:∫
537:∞
532:∞
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479:∞
474:∞
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467:∫
432:ϵ
428:π
349:∞
344:∞
341:−
337:∫
150:integrals
131:Giambiagi
2796:13148774
1647:, where
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884:, where
251:fractals
139:'t Hooft
2946:Bibcode
2926:1701608
2876:Bibcode
2667:Bibcode
2618:Bibcode
1489:Example
904:is the
184:−
143:Veltman
135:Bollini
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400:where
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2792:S2CID
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2972:ISSN
2912:ISBN
2845:link
2827:OCLC
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2737:ISBN
2693:ISSN
2636:ISSN
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145:for
141:and
133:and
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408:A
383:2
379:y
375:+
370:2
366:x
360:y
357:d
333:A
330:=
327:)
324:x
321:(
318:V
298:x
278:s
243:d
239:d
220:d
216:d
208:d
199:i
195:x
190:j
186:x
181:i
177:x
173:d
162:d
110:e
103:t
96:v
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