C-theorem: Difference between revisions

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==Four-dimensional case: ''A''-theorem==

Until[[John recently,Cardy]] itin had1988 notconsidered beenthe possiblepossibility to prove an analoggeneralise ''C''-theorem into higher-dimensional quantum field theory. ItHe is known that at fixed points of the RG flow, if such function exists, it will no more be equal to the central charge {{mvar|c}}, but rather to a different quantity {{mvar|a}}.conjectured<ref>{{cite journal | last1 = NakayamaCardy | first1 = YJohn | year = 20151988 | title = ScaleIs invariancethere vsa conformalc-theorem invariancein four dimensions? | url = | journal = Physics ReportsLetters B | volume = 569215 | issue = | pages = 1–93749 | doi=10.1016/j.physrep.2014.12.0030370-2693(88)90054-8|bibcode = 2015PhR...569....1N }}</ref> Forthat thisin reasonfour spacetime dimensions, the analogquantity ofbehaving monotonically under renormalization group flows, and thus playing the ''C''-theoremrole analogous to the central charge {{mvar|c}} in fourtwo dimensions, is calleda thecertain '''''A''-theorem'''anomaly coefficient which came to be denoted as {{mvar|a}}.

For this reason, the analog of the ''C''-theorem in four dimensions is called the '''''A''-theorem'''.

In perturbation theory, that is for renormalization flows which do not deviate much from free theories, the ''A''-theorem in four dimensions was proved by [[Hugh Osborn]] <ref>{{cite journal | last1 = Osborn | first1 = Hugh | year = 1989 | title = Derivation of a Four-Dimensional c Theorem | url = | journal = Physics Letters B | volume = 222 | issue = | pages = 97 | doi=10.1016/0370-2693(89)90729-6|bibcode = }}
{{cite journal | last1 = Ian | first1 = Jack| last2 = Osborn | first2 = Hugh | year = 1990 | title = Analogs for the c Theorem for Four-Dimensional Renormalizable Field Theories | url = | journal = Nuclear Physics B | volume = 343 | issue = | pages = 647 | doi=10.1016/0550-3213(90)90584-Z|bibcode = }}
</ref> using the local renormalization group equation. However, the problem of finding a proof valid beyond perturbation theory remained open for many years.

In 2011, Zohar Komargodski and Adam Schwimmer of the [[Weizmann Institute of Science]] proposed a nonperturbative proof for the ''A''-theorem, which has gained acceptance.<ref>{{Cite journal | last1 = Reich | first1 = E. S. | doi = 10.1038/nature.2011.9352 | title = Proof found for unifying quantum principle | journal = Nature | year = 2011 | pmid = | pmc = }}</ref><ref name="komargodski">{{Cite journal | last1 = Komargodski | first1 = Z. | last2 = Schwimmer | first2 = A. | doi = 10.1007/JHEP12(2011)099 | title = On renormalization group flows in four dimensions | journal = Journal of High Energy Physics | volume = 2011 | issue = 12 | year = 2011 | pmid = | pmc = |arxiv = 1107.3987 |bibcode = 2011JHEP...12..099K }}</ref> (Still, simultaneous monotonic and cyclic ([[limit cycle]]) or even chaotic RG flows are compatible with such flow functions when multivalued in the couplings, as evinced in specific systems.<ref>{{Cite journal | last1 = Curtright | first1 = T. | last2 = Jin | first2 = X. | last3 = Zachos | first3 = C. | title = Renormalization Group Flows, Cycles, and c-Theorem Folklore | doi = 10.1103/PhysRevLett.108.131601 | journal = Physical Review Letters | volume = 108 | issue = 13 | year = 2012 | pmid = 22540692| pmc = |arxiv = 1111.2649 |bibcode = 2012PhRvL.108m1601C | page=131601}}</ref>) RG flows of theories in 4 dimensions and the question of whether scale invariance implies conformal invariance, is a field of active research and not all questions are settled.

==See also==