C-theorem

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

• ${\displaystyle C(g_{i}^{},\mu )}$ 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 ${\displaystyle g_{i}^{*}}$, the function ${\displaystyle C(g_{i}^{*},\mu )=C_{*}}$ is a constant, independent of energy scale.

If such a so-called C-function exists for a given quantum field theory, it specifies that the RG flow of the theory is irreversible.^[1]

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,^[2] 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 proposed a proof for the physically more important four-dimensional case, which has gained acceptance.^[3][4] 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.^[5]

See also

• Conformal field theory

References

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2. Zamolodchikov, A. B. (1986). "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory, JETP Lett 43, pp 730-732.
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