Until recently, it had not been possible to prove an analog ''C''-theorem in higher-dimensional quantum field theory. <!--{{Citation needed}} begin-->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 c, but rather to a different quantity a.{{citation needed|reason=This claim needs a reliable source.|date=January 2015}}<!--{{Citation needed}} end--> 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 doi| 10.1038/nature.2011.9352|noedit}}</ref><ref name="komargodski">{{citeCite journal doi| last1 = Komargodski | first1 = Z. | last2 = Schwimmer | first2 = A. | doi = 10.1007/JHEP12(2011)099 |noedit 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>{{citeCite journal doi| 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 |noedit journal = Physical Review Letters | volume = 108 | issue = 13 | year = 2012 | pmid = | pmc = |arxiv = 1111.2649 |bibcode = 2012PhRvL.108m1601C }}</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 (circa 2013).
