1525:, functional renormalization does not make a strict distinction between renormalizable and nonrenormalizable couplings. All running couplings that are allowed by symmetries of the problem are generated during the FRG flow. However, the nonrenormalizable couplings approach partial fixed points very quickly during the evolution towards the infrared, and thus the flow effectively collapses on a hypersurface of the dimension given by the number of renormalizable couplings. Taking the nonrenormalizable couplings into account allows to study nonuniversal features that are sensitive to the concrete choice of the microscopic action
43:. This technique allows to interpolate smoothly between the known microscopic laws and the complicated macroscopic phenomena in physical systems. In this sense, it bridges the transition from simplicity of microphysics to complexity of macrophysics. Figuratively speaking, FRG acts as a microscope with a variable resolution. One starts with a high-resolution picture of the known microphysical laws and subsequently decreases the resolution to obtain a coarse-grained picture of macroscopic collective phenomena. The method is nonperturbative, meaning that it does not rely on an expansion in a small
2555:
2796:
2300:
1454:
estimate is nontrivial in functional renormalization. One way to estimate the error in FRG is to improve the truncation in successive steps, i.e. to enlarge the sub-theory space by including more and more running couplings. The difference in the flows for different truncations gives a good estimate of the error. Alternatively, one can use different regulator functions
1585:. While the microscopic theory is defined in terms of two-component nonrelativistic fermions, at low energies a composite (particle-particle) dimer becomes an additional degree of freedom, and it is advisable to include it explicitly in the model. The low-energy composite degrees of freedom can be introduced in the description by the method of partial bosonization (
2309:
1201:
2564:
2097:
1517:
of the liquid–gas phase transition in water and the ferromagnetic phase transition in magnets are the same. In the renormalization group language, different systems from the same universality class flow to the same (partially) stable infrared fixed point. In this way macrophysics becomes independent
1435:
is performed, which is then truncated at finite order leading to a finite system of ordinary differential equations. Different systematic expansion schemes (such as the derivative expansion, vertex expansion, etc.) were developed. The choice of the suitable scheme should be physically motivated and
1453:
The
Wetterich flow equation is an exact equation. However, in practice, the functional differential equation must be truncated, i.e. it must be projected to functions of a few variables or even onto some finite-dimensional sub-theory space. As in every nonperturbative method, the question of error
2550:{\displaystyle {\partial _{\Lambda }}{{\mathcal {W}}_{\Lambda }}=-{\Delta _{{{\dot {D}}_{\Lambda }}+{{\dot {G}}_{0,\Lambda }}}}{{\mathcal {W}}_{\Lambda }}+{e^{-\Delta _{D_{\Lambda }}^{12}}}\Delta _{{\dot {G}}_{0,\Lambda }}^{12}{\mathcal {W}}_{\Lambda }^{(1)}{\mathcal {W}}_{\Lambda }^{(2)}}
1572:
of the
Polchinski functional equation, derived by Joseph Polchinski in 1984. The concept of the effective average action, used in FRG, is, however, more intuitive than the flowing bare action in the Polchinski equation. In addition, the FRG method proved to be more suitable for practical
2791:{\displaystyle \Delta _{{\dot {G}}_{0,\Lambda }}^{12}{\mathcal {V}}_{\Lambda }^{(1)}{\mathcal {V}}_{\Lambda }^{(2)}={\frac {1}{2}}\left({{\frac {\delta {{V}_{\Lambda }}(\psi )}{\delta \psi }},{{\dot {G}}_{0,\Lambda }}{\frac {\delta {{V}_{\Lambda }}(\psi )}{\delta \psi }}}\right)}
604:
1481:
in a given (fixed) truncation and determine the difference of the RG flows in the infrared for the respective regulator choices. If bosonization is used, one can check the insensitivity of final results with respect to different bosonization
2295:{\displaystyle {\frac {\partial }{\partial \Lambda }}{{V}_{\Lambda }}(\psi )=-{{\dot {\Delta }}_{G_{0,\Lambda }}}{{V}_{\Lambda }}(\psi )+\Delta _{{\dot {G}}_{0,\Lambda }}^{12}{\mathcal {V}}_{\Lambda }^{(1)}{\mathcal {V}}_{\Lambda }^{(2)}}
734:
from the left-hand-side and the right-hand-side respectively, due to the tensor structure of the equation. This feature is often shown simplified by the second derivative of the effective action. The functional differential equation for
1609:. In FRG a more efficient way to incorporate macroscopic degrees of freedom was introduced, which is known as flowing bosonization or rebosonization. With the help of a scale-dependent field transformation, this allows to perform the
1576:
Typically, low-energy physics of strongly interacting systems is described by macroscopic degrees of freedom (i.e. particle excitations) which are very different from microscopic high-energy degrees of freedom. For instance,
1443:
Note however, that due to multiple choices regarding (prefactor-)conventions and the concrete definition of the effective action, one can find other (equivalent) versions of the
Wetterich equation in the literature.
2066:
1997:
1798:
465:
1040:
799:
916:
164:. Interesting physics, as propagators and effective couplings for interactions, can be straightforwardly extracted from it. In a generic interacting field theory the effective action
1195:
1581:
is a field theory of interacting quarks and gluons. At low energies, however, proper degrees of freedom are baryons and mesons. Another example is the BEC/BCS crossover problem in
1376:
705:
943:
denotes a supertrace which sums over momenta, frequencies, internal indices, and fields (taking bosons with a plus and fermions with a minus sign). The exact flow equation for
325:
641:
941:
1433:
1253:
1097:
968:
760:
732:
458:
405:
378:
236:
1156:
845:
431:
35:(RG) concept which is used in quantum and statistical field theory, especially when dealing with strongly interacting systems. The method combines functional methods of
1130:
2090:
1607:
1563:
202:
182:
158:
130:
84:
2947:
In gauge quantum field theory, FRG was used, for instance, to investigate the chiral phase transition and infrared properties of QCD and its large-flavor extensions.
1507:
874:
1827:
1479:
1403:
1311:
1280:
1067:
352:
287:
2844:
2942:
2916:
2890:
1337:
2864:
1892:
1869:
1631:
1543:
1226:
819:
661:
256:
110:
1513:. Universality manifests itself in the observation that some very distinct physical systems have the same critical behavior. For instance, to good accuracy,
2072:
and excludes from the interaction all possible terms, formed by a convolution of source fields with respective Green function D. Introducing some cutoff
3462:
M. Reuter and F. Saueressig; Frank
Saueressig (2007). "Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity".
1485:
In FRG, as in all RG methods, a lot of insight about a physical system can be gained from the topology of RG flows. Specifically, identification of
1610:
1586:
3410:
3363:
3006:
2005:
3501:
3240:
J. Berges; N. Tetradis; C. Wetterich (2002), "Non-perturbative renormalization flow in quantum field theory and statistical mechanics",
1489:
of the renormalization group evolution is of great importance. Near fixed points the flow of running couplings effectively stops and RG
1899:
3220:
1650:
1510:
599:{\displaystyle k\,\partial _{k}\Gamma _{k}={\frac {1}{2}}{\text{STr}}\,k\,\partial _{k}R_{k}\,(\Gamma _{k}^{(1,1)}+R_{k})^{-1},}
3491:
3486:
977:
1509:-functions approach zero. The presence of (partially) stable infrared fixed points is closely connected to the concept of
3496:
1408:
In most cases of interest the
Wetterich equation can only be solved approximately. Usually some type of expansion of
1132:
allowed by the symmetries of the problem. As schematically shown in the figure, at the microscopic ultraviolet scale
1486:
2951:
2810:
1582:
765:
259:
3430:
Salmhofer, Manfred; Honerkamp, Carsten (2001), "Fermionic renormalization group flows: Technique and theory",
882:
1436:
depends on a given problem. The expansions do not necessarily involve a small parameter (like an interaction
1099:
can be illustrated in the theory space, which is a multi-dimensional space of all possible running couplings
1569:
1161:
112:
and depends on the fields of a given theory. It includes all quantum and thermal fluctuations. Variation of
1643:
1342:
3173:
1578:
666:
2986:
1283:
205:
32:
292:
3439:
3307:
3259:
3165:
3125:
3089:
3043:
60:
36:
3178:
2958:
or frustrated magnetic systems), repulsive Bose gas, BEC/BCS crossover for two-component Fermi gas,
848:
1522:
1255:
evolves in the theory space according to the functional flow equation. The choice of the regulator
971:
619:
613:
20:
2965:
Application of FRG to gravity provided arguments in favor of nonperturbative renormalizability of
924:
47:. Mathematically, FRG is based on an exact functional differential equation for a scale-dependent
3463:
3416:
3388:
3369:
3341:
3323:
3297:
3275:
3249:
3191:
3155:
3105:
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2996:
1514:
1411:
1231:
1075:
946:
738:
710:
609:
436:
383:
357:
214:
1135:
824:
410:
1204:
Renormalization group flow in the theory space of all possible couplings allowed by symmetries.
1102:
3406:
3359:
3216:
2970:
2814:
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1592:
1548:
1437:
87:
44:
40:
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115:
69:
3447:
3398:
3351:
3315:
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3133:
3097:
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3001:
2976:
In mathematical physics FRG was used to prove renormalizability of different field theories.
1492:
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90:
64:
48:
1805:
1457:
1381:
1289:
1258:
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265:
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1642:
Contrary to the flow equation for the effective action, this scheme is formulated for the
161:
137:
2921:
2895:
2869:
1316:
3443:
3311:
3263:
3169:
3129:
3093:
3047:
3336:
H.Gies (2006). "Introduction to the functional RG and applications to gauge theories".
2849:
1877:
1832:
1616:
1528:
1211:
804:
646:
241:
95:
3271:
238:
often called average action or flowing action. The dependence on the RG sliding scale
3480:
3383:
B. Delamotte (2007). "An introduction to the nonperturbative renormalization group".
3288:
J. Polonyi, Janos (2003), "Lectures on the functional renormalization group method",
3279:
3195:
3137:
3063:
3055:
2955:
3420:
3373:
3327:
3109:
3070:
Morris, T. R. (1994), "The Exact renormalization group and approximate solutions",
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2069:
1802:
which generates n-particle interaction vertices, amputated by the bare propagators
3402:
3385:
Renormalization Group and
Effective Field Theory Approaches to Many-Body Systems
3355:
3338:
Renormalization Group and
Effective Field Theory Approaches to Many-Body Systems
1042:
is the full inverse field propagator modified by the presence of the regulator
974:, where multi-loop diagrams must be included. The second functional derivative
380:
by giving them a large mass, while high momentum modes are not affected. Thus,
3101:
3024:
Wetterich, C. (1993), "Exact evolution equation for the effective potential",
184:, however, is difficult to obtain. FRG provides a practical tool to calculate
3146:
Reuter, M. (1998), "Nonperturbative evolution equation for quantum gravity",
3187:
1871:
is the "standard" generating functional for the n-particle Green functions.
133:
211:
The central object in FRG is a scale-dependent effective action functional
1874:
The Wick ordering of effective interaction with respect to Green function
1589:). This transformation, however, is done once and for all at the UV scale
970:
has a one-loop structure. This is an important simplification compared to
3393:
1200:
3346:
3319:
3302:
3254:
3160:
3084:
2954:, the method proved to be successful to treat lattice models (e.g. the
1638:
Functional renormalization-group for Wick-ordered effective interaction
1313:
correspond to the different paths in the figure. At the infrared scale
3452:
3116:
Polchinski, J. (1984), "Renormalization and
Effective Lagrangians",
3038:
2061:{\displaystyle \Delta =D\delta ^{2}/(\delta \eta \delta \eta ^{+})}
1405:, and all trajectories meet at the same point in the theory space.
3468:
2068:
is the
Laplacian in the field space. This operation is similar to
1199:
1282:
is not unique, which introduces some scheme dependence into the
2805:
The method was applied to numerous problems in physics, e.g.:
1992:{\displaystyle {\mathcal {W}}=\exp(-\Delta _{D}){\mathcal {V}}}
407:
includes all quantum and statistical fluctuations with momenta
1793:{\displaystyle {\mathcal {V}}=-\ln Z-\eta G_{0}^{-1}\eta ^{+}}
3340:. Lecture Notes in Physics. Vol. 852. pp. 287–348.
2892:
and the
Berezinskii–Kosterlitz–Thouless phase transition for
160:
is the generating functional of the one-particle irreducible
3387:. Lecture Notes in Physics. Vol. 852. pp. 49–132.
2633:
2608:
2525:
2500:
2411:
2329:
2270:
2245:
1962:
1905:
1656:
1518:
of the microscopic details of the particular physical model.
1440:) and thus they are, in general, of nonperturbative nature.
821:
describes the physics at the microscopic ultraviolet scale
3211:
Kopietz, Peter; Bartosch, Lorenz; SchĂĽtz, Florian (2010).
1286:
flow. For this reason, different choices of the regulator
132:
yields exact quantum field equations, for example for
2924:
2898:
2872:
2852:
2823:
2567:
2312:
2100:
2078:
2008:
1902:
1880:
1835:
1808:
1653:
1619:
1595:
1551:
1531:
1495:
1460:
1414:
1384:
1345:
1319:
1292:
1261:
1234:
1214:
1164:
1138:
1105:
1078:
1048:
1035:{\displaystyle \Gamma _{k}^{(2)}=\Gamma _{k}^{(1,1)}}
980:
949:
927:
885:
856:
827:
807:
768:
741:
713:
669:
649:
622:
468:
439:
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386:
360:
333:
295:
268:
244:
217:
190:
170:
146:
118:
98:
72:
3213:
Introduction to the Functional Renormalization Group
2846:-symmetric scalar theories in different dimensions
2962:, disordered systems and nonequilibrium phenomena.
2936:
2910:
2884:
2858:
2838:
2790:
2549:
2294:
2084:
2060:
1991:
1886:
1863:
1821:
1792:
1625:
1601:
1557:
1537:
1501:
1473:
1427:
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1189:
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1091:
1061:
1034:
962:
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910:
868:
839:
813:
793:
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655:
643:denotes a derivative with respect to the RG scale
635:
598:
452:
425:
399:
372:
346:
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281:
250:
230:
196:
176:
152:
124:
104:
78:
39:with the intuitive renormalization group idea of
1568:The Wetterich equation can be obtained from the
762:must be supplemented with the initial condition
1378:is recovered for every choice of the cut-off
8:
2304:takes the form of the Wick-ordered equation
1119:
1106:
663:at fixed values of the fields. Furthermore,
2969:in four spacetime dimensions, known as the
16:Implementation of the renormalization group
55:The flow equation for the effective action
3467:
3451:
3392:
3345:
3301:
3253:
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3159:
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2510:
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2250:
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2211:
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2148:
2147:
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2123:
2118:
2116:
2101:
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2077:
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2022:
2007:
1980:
1961:
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1951:
1923:
1904:
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1879:
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1834:
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1807:
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1169:
1163:
1137:
1113:
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1077:
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1047:
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1009:
990:
985:
979:
954:
948:
928:
926:
896:
884:
855:
826:
806:
794:{\displaystyle \Gamma _{k\to \Lambda }=S}
773:
767:
746:
740:
718:
712:
679:
674:
668:
648:
627:
621:
584:
574:
549:
544:
536:
530:
520:
515:
511:
506:
496:
487:
477:
472:
467:
460:obeys the exact functional flow equation
444:
438:
412:
391:
385:
359:
338:
332:
305:
300:
294:
273:
267:
243:
222:
216:
189:
169:
145:
117:
97:
71:
911:{\displaystyle \Gamma =\Gamma _{k\to 0}}
3203:
1072:The renormalization group evolution of
1190:{\displaystyle \Gamma _{k=\Lambda }=S}
1158:one starts with the initial condition
1448:Aspects of functional renormalization
1371:{\displaystyle \Gamma _{k=0}=\Gamma }
1339:, however, the full effective action
707:denotes the functional derivative of
7:
3007:Asymptotic safety in quantum gravity
2813:, FRG provided a unified picture of
140:of superconductors. Mathematically,
2866:, including critical exponents for
1611:Hubbard–Stratonovich transformation
1587:Hubbard–Stratonovich transformation
700:{\displaystyle \Gamma _{k}^{(1,1)}}
2757:
2737:
2687:
2639:
2614:
2593:
2569:
2531:
2506:
2485:
2461:
2445:
2436:
2417:
2398:
2368:
2349:
2335:
2319:
2315:
2276:
2251:
2230:
2206:
2187:
2171:
2150:
2124:
2110:
2107:
2103:
2079:
2009:
1948:
1596:
1552:
1545:and the finite ultraviolet cutoff
1416:
1365:
1347:
1236:
1176:
1166:
1145:
1080:
1006:
982:
951:
893:
886:
834:
780:
770:
743:
715:
671:
624:
541:
517:
484:
474:
441:
388:
354:decouples slow modes with momenta
327:. Roughly speaking, the regulator
297:
219:
191:
171:
147:
119:
73:
14:
320:{\displaystyle \Gamma _{k}^{(2)}}
25:functional renormalization group
1228:is lowered, the flowing action
801:, where the "classical action"
289:to the full inverse propagator
2833:
2827:
2769:
2763:
2699:
2693:
2650:
2644:
2625:
2619:
2542:
2536:
2517:
2511:
2287:
2281:
2262:
2256:
2199:
2193:
2136:
2130:
2055:
2033:
1986:
1967:
1957:
1941:
1929:
1910:
1858:
1839:
1753:
1698:
1680:
1661:
1613:continuously at all RG scales
1027:
1015:
997:
991:
900:
860:
777:
692:
680:
581:
562:
550:
537:
312:
306:
31:) is an implementation of the
1:
3272:10.1016/S0370-1573(01)00098-9
636:{\displaystyle \partial _{k}}
3138:10.1016/0550-3213(84)90287-6
3056:10.1016/0370-2693(93)90726-X
936:{\displaystyle {\text{STr}}}
3403:10.1007/978-3-642-27320-9_2
3356:10.1007/978-3-642-27320-9_6
1428:{\displaystyle \Gamma _{k}}
1248:{\displaystyle \Gamma _{k}}
1092:{\displaystyle \Gamma _{k}}
963:{\displaystyle \Gamma _{k}}
755:{\displaystyle \Gamma _{k}}
727:{\displaystyle \Gamma _{k}}
453:{\displaystyle \Gamma _{k}}
400:{\displaystyle \Gamma _{k}}
373:{\displaystyle q\lesssim k}
231:{\displaystyle \Gamma _{k}}
3518:
3502:Fixed points (mathematics)
1151:{\displaystyle k=\Lambda }
840:{\displaystyle k=\Lambda }
426:{\displaystyle q\gtrsim k}
258:is introduced by adding a
3102:10.1142/S0217751X94000972
2092:the Polchinskii equation
1125:{\displaystyle \{c_{n}\}}
2952:condensed matter physics
2811:statistical field theory
2085:{\displaystyle \Lambda }
1602:{\displaystyle \Lambda }
1583:condensed matter physics
1558:{\displaystyle \Lambda }
3188:10.1103/PhysRevD.57.971
1570:Legendre transformation
197:{\displaystyle \Gamma }
177:{\displaystyle \Gamma }
153:{\displaystyle \Gamma }
125:{\displaystyle \Gamma }
79:{\displaystyle \Gamma }
2938:
2912:
2886:
2860:
2840:
2792:
2551:
2296:
2086:
2062:
1993:
1888:
1865:
1823:
1794:
1627:
1603:
1579:quantum chromodynamics
1559:
1539:
1503:
1502:{\displaystyle \beta }
1475:
1429:
1399:
1372:
1333:
1307:
1276:
1249:
1222:
1205:
1191:
1152:
1126:
1093:
1063:
1036:
964:
937:
912:
870:
869:{\displaystyle k\to 0}
847:. Importantly, in the
841:
815:
795:
756:
728:
701:
657:
637:
600:
454:
427:
401:
374:
348:
321:
283:
252:
232:
198:
178:
154:
126:
106:
86:is an analogue of the
80:
3492:Renormalization group
3487:Statistical mechanics
2987:Renormalization group
2939:
2913:
2887:
2861:
2841:
2793:
2552:
2297:
2087:
2063:
1994:
1889:
1866:
1824:
1822:{\displaystyle G_{0}}
1795:
1644:effective interaction
1628:
1604:
1560:
1540:
1504:
1476:
1474:{\displaystyle R_{k}}
1430:
1400:
1398:{\displaystyle R_{k}}
1373:
1334:
1308:
1306:{\displaystyle R_{k}}
1284:renormalization group
1277:
1275:{\displaystyle R_{k}}
1250:
1223:
1208:As the sliding scale
1203:
1192:
1153:
1127:
1094:
1064:
1062:{\displaystyle R_{k}}
1037:
965:
938:
913:
871:
842:
816:
796:
757:
729:
702:
658:
638:
601:
455:
433:. The flowing action
428:
402:
375:
349:
347:{\displaystyle R_{k}}
322:
284:
282:{\displaystyle R_{k}}
253:
233:
206:renormalization group
199:
179:
155:
127:
107:
81:
33:renormalization group
3072:Int. J. Mod. Phys. A
2922:
2896:
2870:
2850:
2839:{\displaystyle O(N)}
2821:
2817:in classical linear
2565:
2310:
2098:
2076:
2006:
1900:
1878:
1833:
1806:
1651:
1617:
1593:
1549:
1529:
1493:
1458:
1412:
1382:
1343:
1317:
1290:
1259:
1232:
1212:
1162:
1136:
1103:
1076:
1046:
978:
947:
925:
918:is obtained. In the
883:
854:
825:
805:
766:
739:
711:
667:
647:
620:
466:
437:
411:
384:
358:
331:
293:
266:
242:
215:
188:
168:
144:
116:
96:
70:
61:quantum field theory
37:quantum field theory
3444:2001PThPh.105....1S
3312:2003CEJPh...1....1P
3290:Cent. Eur. J. Phys.
3264:2002PhR...363..223B
3170:1998PhRvD..57..971R
3130:1984NuPhB.231..269P
3094:1994IJMPA...9.2411M
3048:1993PhLB..301...90W
2937:{\displaystyle N=2}
2911:{\displaystyle d=2}
2885:{\displaystyle d=3}
2654:
2629:
2604:
2546:
2521:
2496:
2456:
2291:
2266:
2241:
1779:
1742:
1718:
1523:perturbation theory
1332:{\displaystyle k=0}
1031:
1001:
972:perturbation theory
696:
566:
316:
21:theoretical physics
3497:Scaling symmetries
3432:Prog. Theor. Phys.
3320:10.2478/BF02475552
2997:Critical phenomena
2934:
2908:
2882:
2856:
2836:
2788:
2630:
2605:
2568:
2547:
2522:
2497:
2460:
2435:
2292:
2267:
2242:
2205:
2082:
2058:
1989:
1894:can be defined by
1884:
1861:
1819:
1790:
1762:
1725:
1701:
1623:
1599:
1555:
1535:
1515:critical exponents
1499:
1471:
1425:
1395:
1368:
1329:
1303:
1272:
1245:
1218:
1206:
1187:
1148:
1122:
1089:
1059:
1032:
1005:
981:
960:
933:
920:Wetterich equation
908:
866:
837:
811:
791:
752:
724:
697:
670:
653:
633:
610:Christof Wetterich
596:
540:
450:
423:
397:
370:
344:
317:
296:
279:
262:(infrared cutoff)
248:
228:
194:
174:
150:
122:
102:
76:
3453:10.1143/PTP.105.1
3412:978-3-642-27319-3
3365:978-3-642-27319-3
3234:Pedagogic reviews
3078:(14): 2411–2449,
2971:asymptotic safety
2859:{\displaystyle d}
2815:phase transitions
2781:
2727:
2711:
2666:
2583:
2475:
2388:
2364:
2220:
2156:
2114:
1887:{\displaystyle D}
1864:{\displaystyle Z}
1626:{\displaystyle k}
1538:{\displaystyle S}
1438:coupling constant
1221:{\displaystyle k}
931:
814:{\displaystyle S}
656:{\displaystyle k}
509:
504:
251:{\displaystyle k}
105:{\displaystyle S}
91:action functional
45:coupling constant
41:Kenneth G. Wilson
3509:
3473:
3471:
3456:
3455:
3424:
3396:
3394:cond-mat/0702365
3377:
3349:
3330:
3305:
3282:
3257:
3248:(4–6): 223–386,
3227:
3226:
3208:
3198:
3181:
3163:
3140:
3112:
3087:
3066:
3041:
3002:Scale invariance
2943:
2941:
2940:
2935:
2917:
2915:
2914:
2909:
2891:
2889:
2888:
2883:
2865:
2863:
2862:
2857:
2845:
2843:
2842:
2837:
2797:
2795:
2794:
2789:
2787:
2783:
2782:
2780:
2772:
2762:
2761:
2760:
2755:
2744:
2742:
2741:
2740:
2729:
2728:
2720:
2712:
2710:
2702:
2692:
2691:
2690:
2685:
2674:
2667:
2659:
2653:
2642:
2637:
2636:
2628:
2617:
2612:
2611:
2603:
2598:
2597:
2596:
2585:
2584:
2576:
2556:
2554:
2553:
2548:
2545:
2534:
2529:
2528:
2520:
2509:
2504:
2503:
2495:
2490:
2489:
2488:
2477:
2476:
2468:
2459:
2458:
2457:
2455:
2450:
2449:
2448:
2422:
2421:
2420:
2415:
2414:
2406:
2405:
2404:
2403:
2402:
2401:
2390:
2389:
2381:
2373:
2372:
2371:
2366:
2365:
2357:
2340:
2339:
2338:
2333:
2332:
2324:
2323:
2322:
2301:
2299:
2298:
2293:
2290:
2279:
2274:
2273:
2265:
2254:
2249:
2248:
2240:
2235:
2234:
2233:
2222:
2221:
2213:
2192:
2191:
2190:
2185:
2178:
2177:
2176:
2175:
2174:
2158:
2157:
2149:
2129:
2128:
2127:
2122:
2115:
2113:
2102:
2091:
2089:
2088:
2083:
2067:
2065:
2064:
2059:
2054:
2053:
2032:
2027:
2026:
1998:
1996:
1995:
1990:
1985:
1984:
1966:
1965:
1956:
1955:
1928:
1927:
1909:
1908:
1893:
1891:
1890:
1885:
1870:
1868:
1867:
1862:
1857:
1856:
1828:
1826:
1825:
1820:
1818:
1817:
1799:
1797:
1796:
1791:
1789:
1788:
1778:
1770:
1752:
1751:
1741:
1733:
1717:
1709:
1679:
1678:
1660:
1659:
1632:
1630:
1629:
1624:
1608:
1606:
1605:
1600:
1564:
1562:
1561:
1556:
1544:
1542:
1541:
1536:
1521:Compared to the
1508:
1506:
1505:
1500:
1480:
1478:
1477:
1472:
1470:
1469:
1434:
1432:
1431:
1426:
1424:
1423:
1404:
1402:
1401:
1396:
1394:
1393:
1377:
1375:
1374:
1369:
1361:
1360:
1338:
1336:
1335:
1330:
1312:
1310:
1309:
1304:
1302:
1301:
1281:
1279:
1278:
1273:
1271:
1270:
1254:
1252:
1251:
1246:
1244:
1243:
1227:
1225:
1224:
1219:
1196:
1194:
1193:
1188:
1180:
1179:
1157:
1155:
1154:
1149:
1131:
1129:
1128:
1123:
1118:
1117:
1098:
1096:
1095:
1090:
1088:
1087:
1068:
1066:
1065:
1060:
1058:
1057:
1041:
1039:
1038:
1033:
1030:
1013:
1000:
989:
969:
967:
966:
961:
959:
958:
942:
940:
939:
934:
932:
929:
917:
915:
914:
909:
907:
906:
878:effective action
875:
873:
872:
867:
846:
844:
843:
838:
820:
818:
817:
812:
800:
798:
797:
792:
784:
783:
761:
759:
758:
753:
751:
750:
733:
731:
730:
725:
723:
722:
706:
704:
703:
698:
695:
678:
662:
660:
659:
654:
642:
640:
639:
634:
632:
631:
605:
603:
602:
597:
592:
591:
579:
578:
565:
548:
535:
534:
525:
524:
510:
507:
505:
497:
492:
491:
482:
481:
459:
457:
456:
451:
449:
448:
432:
430:
429:
424:
406:
404:
403:
398:
396:
395:
379:
377:
376:
371:
353:
351:
350:
345:
343:
342:
326:
324:
323:
318:
315:
304:
288:
286:
285:
280:
278:
277:
257:
255:
254:
249:
237:
235:
234:
229:
227:
226:
203:
201:
200:
195:
183:
181:
180:
175:
162:Feynman diagrams
159:
157:
156:
151:
131:
129:
128:
123:
111:
109:
108:
103:
85:
83:
82:
77:
65:effective action
49:effective action
3517:
3516:
3512:
3511:
3510:
3508:
3507:
3506:
3477:
3476:
3461:
3429:
3413:
3382:
3366:
3335:
3287:
3239:
3236:
3231:
3230:
3223:
3210:
3209:
3205:
3179:10.1.1.263.3439
3145:
3115:
3069:
3023:
3020:
3015:
2992:Renormalization
2983:
2967:quantum gravity
2920:
2919:
2894:
2893:
2868:
2867:
2848:
2847:
2819:
2818:
2803:
2773:
2750:
2745:
2717:
2703:
2680:
2675:
2668:
2573:
2563:
2562:
2465:
2440:
2427:
2408:
2378:
2354:
2348:
2326:
2314:
2308:
2307:
2210:
2180:
2160:
2146:
2117:
2106:
2096:
2095:
2074:
2073:
2045:
2018:
2004:
2003:
1976:
1947:
1919:
1898:
1897:
1876:
1875:
1848:
1831:
1830:
1809:
1804:
1803:
1780:
1743:
1670:
1649:
1648:
1640:
1615:
1614:
1591:
1590:
1547:
1546:
1527:
1526:
1491:
1490:
1461:
1456:
1455:
1450:
1415:
1410:
1409:
1385:
1380:
1379:
1346:
1341:
1340:
1315:
1314:
1293:
1288:
1287:
1262:
1257:
1256:
1235:
1230:
1229:
1210:
1209:
1165:
1160:
1159:
1134:
1133:
1109:
1101:
1100:
1079:
1074:
1073:
1049:
1044:
1043:
976:
975:
950:
945:
944:
923:
922:
892:
881:
880:
852:
851:
823:
822:
803:
802:
769:
764:
763:
742:
737:
736:
714:
709:
708:
665:
664:
645:
644:
623:
618:
617:
580:
570:
526:
516:
483:
473:
464:
463:
440:
435:
434:
409:
408:
387:
382:
381:
356:
355:
334:
329:
328:
291:
290:
269:
264:
263:
240:
239:
218:
213:
212:
186:
185:
166:
165:
142:
141:
138:electrodynamics
114:
113:
94:
93:
68:
67:
57:
17:
12:
11:
5:
3515:
3513:
3505:
3504:
3499:
3494:
3489:
3479:
3478:
3475:
3474:
3458:
3457:
3426:
3425:
3411:
3379:
3378:
3364:
3347:hep-ph/0611146
3332:
3331:
3303:hep-th/0110026
3284:
3283:
3255:hep-ph/0005122
3235:
3232:
3229:
3228:
3221:
3202:
3201:
3200:
3199:
3161:hep-th/9605030
3154:(2): 971–985,
3142:
3141:
3113:
3085:hep-ph/9308265
3067:
3019:
3016:
3014:
3011:
3010:
3009:
3004:
2999:
2994:
2989:
2982:
2979:
2978:
2977:
2974:
2963:
2948:
2945:
2933:
2930:
2927:
2907:
2904:
2901:
2881:
2878:
2875:
2855:
2835:
2832:
2829:
2826:
2802:
2799:
2786:
2779:
2776:
2771:
2768:
2765:
2759:
2754:
2748:
2739:
2736:
2733:
2726:
2723:
2715:
2709:
2706:
2701:
2698:
2695:
2689:
2684:
2678:
2671:
2665:
2662:
2657:
2652:
2649:
2646:
2641:
2635:
2627:
2624:
2621:
2616:
2610:
2602:
2595:
2592:
2589:
2582:
2579:
2571:
2544:
2541:
2538:
2533:
2527:
2519:
2516:
2513:
2508:
2502:
2494:
2487:
2484:
2481:
2474:
2471:
2463:
2454:
2447:
2443:
2438:
2434:
2430:
2425:
2419:
2413:
2400:
2397:
2394:
2387:
2384:
2376:
2370:
2363:
2360:
2351:
2346:
2343:
2337:
2331:
2321:
2317:
2289:
2286:
2283:
2278:
2272:
2264:
2261:
2258:
2253:
2247:
2239:
2232:
2229:
2226:
2219:
2216:
2208:
2204:
2201:
2198:
2195:
2189:
2184:
2173:
2170:
2167:
2163:
2155:
2152:
2144:
2141:
2138:
2135:
2132:
2126:
2121:
2112:
2109:
2105:
2081:
2057:
2052:
2048:
2044:
2041:
2038:
2035:
2031:
2025:
2021:
2017:
2014:
2011:
1988:
1983:
1979:
1975:
1972:
1969:
1964:
1959:
1954:
1950:
1946:
1943:
1940:
1937:
1934:
1931:
1926:
1922:
1918:
1915:
1912:
1907:
1883:
1860:
1855:
1851:
1847:
1844:
1841:
1838:
1816:
1812:
1787:
1783:
1777:
1774:
1769:
1765:
1761:
1758:
1755:
1750:
1746:
1740:
1737:
1732:
1728:
1724:
1721:
1716:
1713:
1708:
1704:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1677:
1673:
1669:
1666:
1663:
1658:
1639:
1636:
1635:
1634:
1622:
1598:
1574:
1566:
1554:
1534:
1519:
1498:
1483:
1468:
1464:
1449:
1446:
1422:
1418:
1392:
1388:
1367:
1364:
1359:
1356:
1353:
1349:
1328:
1325:
1322:
1300:
1296:
1269:
1265:
1242:
1238:
1217:
1186:
1183:
1178:
1175:
1172:
1168:
1147:
1144:
1141:
1121:
1116:
1112:
1108:
1086:
1082:
1056:
1052:
1029:
1026:
1023:
1020:
1017:
1012:
1008:
1004:
999:
996:
993:
988:
984:
957:
953:
905:
902:
899:
895:
891:
888:
865:
862:
859:
849:infrared limit
836:
833:
830:
810:
790:
787:
782:
779:
776:
772:
749:
745:
721:
717:
694:
691:
688:
685:
682:
677:
673:
652:
630:
626:
616:in 1993. Here
595:
590:
587:
583:
577:
573:
569:
564:
561:
558:
555:
552:
547:
543:
539:
533:
529:
523:
519:
514:
503:
500:
495:
490:
486:
480:
476:
471:
447:
443:
422:
419:
416:
394:
390:
369:
366:
363:
341:
337:
314:
311:
308:
303:
299:
276:
272:
247:
225:
221:
204:employing the
193:
173:
149:
121:
101:
75:
56:
53:
15:
13:
10:
9:
6:
4:
3:
2:
3514:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3484:
3482:
3470:
3465:
3460:
3459:
3454:
3449:
3445:
3441:
3437:
3433:
3428:
3427:
3422:
3418:
3414:
3408:
3404:
3400:
3395:
3390:
3386:
3381:
3380:
3375:
3371:
3367:
3361:
3357:
3353:
3348:
3343:
3339:
3334:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3304:
3299:
3295:
3291:
3286:
3285:
3281:
3277:
3273:
3269:
3265:
3261:
3256:
3251:
3247:
3243:
3238:
3237:
3233:
3224:
3222:9783642050947
3218:
3214:
3207:
3204:
3197:
3193:
3189:
3185:
3180:
3175:
3171:
3167:
3162:
3157:
3153:
3149:
3144:
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3118:Nucl. Phys. B
3114:
3111:
3107:
3103:
3099:
3095:
3091:
3086:
3081:
3077:
3073:
3068:
3065:
3061:
3057:
3053:
3049:
3045:
3040:
3035:
3031:
3027:
3026:Phys. Lett. B
3022:
3021:
3017:
3012:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2984:
2980:
2975:
2972:
2968:
2964:
2961:
2957:
2956:Hubbard model
2953:
2949:
2946:
2931:
2928:
2925:
2905:
2902:
2899:
2879:
2876:
2873:
2853:
2830:
2824:
2816:
2812:
2808:
2807:
2806:
2800:
2798:
2784:
2777:
2774:
2766:
2752:
2746:
2734:
2731:
2724:
2721:
2713:
2707:
2704:
2696:
2682:
2676:
2669:
2663:
2660:
2655:
2647:
2622:
2600:
2590:
2587:
2580:
2577:
2560:
2557:
2539:
2514:
2492:
2482:
2479:
2472:
2469:
2452:
2441:
2432:
2428:
2423:
2395:
2392:
2385:
2382:
2374:
2361:
2358:
2344:
2341:
2305:
2302:
2284:
2259:
2237:
2227:
2224:
2217:
2214:
2202:
2196:
2182:
2168:
2165:
2161:
2153:
2142:
2139:
2133:
2119:
2093:
2071:
2050:
2046:
2042:
2039:
2036:
2029:
2023:
2019:
2015:
2012:
2000:
1981:
1977:
1973:
1970:
1952:
1944:
1938:
1935:
1932:
1924:
1920:
1916:
1913:
1895:
1881:
1872:
1853:
1849:
1845:
1842:
1836:
1814:
1810:
1800:
1785:
1781:
1775:
1772:
1767:
1763:
1759:
1756:
1748:
1744:
1738:
1735:
1730:
1726:
1722:
1719:
1714:
1711:
1706:
1702:
1695:
1692:
1689:
1686:
1683:
1675:
1671:
1667:
1664:
1646:
1645:
1637:
1620:
1612:
1588:
1584:
1580:
1575:
1573:calculations.
1571:
1567:
1532:
1524:
1520:
1516:
1512:
1496:
1488:
1484:
1466:
1462:
1452:
1451:
1447:
1445:
1441:
1439:
1420:
1406:
1390:
1386:
1362:
1357:
1354:
1351:
1326:
1323:
1320:
1298:
1294:
1285:
1267:
1263:
1240:
1215:
1202:
1198:
1184:
1181:
1173:
1170:
1142:
1139:
1114:
1110:
1084:
1070:
1054:
1050:
1024:
1021:
1018:
1010:
1002:
994:
986:
973:
955:
921:
903:
897:
889:
879:
863:
857:
850:
831:
828:
808:
788:
785:
774:
747:
719:
689:
686:
683:
675:
650:
628:
615:
614:Tim R. Morris
611:
606:
593:
588:
585:
575:
571:
567:
559:
556:
553:
545:
531:
527:
521:
512:
501:
498:
493:
488:
478:
469:
461:
445:
420:
417:
414:
392:
367:
364:
361:
339:
335:
309:
301:
274:
270:
261:
245:
223:
209:
207:
163:
139:
135:
99:
92:
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66:
62:
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50:
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3215:. Springer.
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3206:
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3148:Phys. Rev. D
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3075:
3071:
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2960:Kondo effect
2804:
2801:Applications
2561:
2558:
2306:
2303:
2094:
2070:Normal order
2001:
1896:
1873:
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1511:universality
1487:fixed points
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3296:(1): 1–71,
1482:procedures.
608:derived by
3481:Categories
3242:Phys. Rep.
3124:(2): 269,
3039:1710.05815
3013:References
3469:0708.1317
3280:119033356
3196:119454616
3174:CiteSeerX
3064:119536989
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2778:ψ
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418:≳
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298:Γ
260:regulator
220:Γ
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120:Γ
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3060:S2CID
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