# C-theorem

In theoretical physics, specifically quantum field theory, Zamolodchikov's C-theorem states that there exists a positive real function, $C(g^{}_i,\mu)$, depending on the coupling constants of the quantum field theory considered, $g^{}_i$, and on the energy scale, $\mu^{}_{}$, which has the following properties:

• $C(g^{}_i,\mu)$ decreases monotonically under the renormalization group (RG) flow.
• At fixed points of the RG flow, which are specified by a set of fixed-point couplings $g^*_i$, the function $C(g^*_i,\mu)=C_*$ is a constant, independent of energy scale.

Alexander Zamolodchikov proved in 1986 that two-dimensional quantum field theory always has such a C-function. Moreover, at fixed points of the RG flow, which correspond to conformal field theories, Zamolodchikov's C-function is equal to the central charge of the corresponding conformal field theory,[1] and roughly counts the degrees of freedom of the system.

Until recently, it had not been possible to prove an analog C-theorem in higher-dimensional quantum field theory. However, in 2011, Zohar Komargodski and Adam Schwimmer of the Weizmann Institute of Science proposed a proof for the physically more important four-dimensional case, which has gained acceptance.[2][3] (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.[4]) 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).