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Until recently, it had not been possible to prove an analog ''C''-theorem in higher-dimensional quantum field theory. It 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}}.<ref>{{cite journal | last1 = Nakayama | first1 = Y | year = 2015 | title = Scale invariance vs conformal invariance | url = | journal = Physics Reports | volume = 569 | issue = | pages = 1–93 | doi=10.1016/j.physrep.2014.12.003|bibcode = 2015PhR...569....1N }}</ref> For this reason, the analog of the ''C''-theorem in four dimensions is called the '''''A''-theorem'''.
In 2011, Zohar Komargodski and Adam Schwimmer of the [[Weizmann Institute of Science]] proposed a 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==
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