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==Four-dimensional case: ''A''-theorem==
[[John Cardy]] in 1988 considered the possibility to generalise ''C''-theorem to higher-dimensional quantum field theory. He conjectured<ref>{{cite journal | last1 = Cardy | first1 = John | year = 1988 | title = Is there a c-theorem in four dimensions? | url = | journal = Physics Letters B | volume = 215 | issue = | pages = 749 | doi=10.1016/0370-2693(88)90054-8|bibcode =1988PhLB..215..749C }}</ref> that in four spacetime dimensions, the quantity behaving monotonically under renormalization group flows, and thus playing the role analogous to the central charge {{mvar|c}} in two dimensions, is a certain 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 =1989PhLB..222...97O }}
{{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 =1990NuPhB.343..647J }}
</ref> using the local renormalization group equation. However, the problem of finding a proof valid beyond perturbation theory remained open for many years.
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