Zamolodchikov c-theorem

Remarks

The theorem was proved by Alexander Zamolodchikov in 1986. In complex coordinates, write $T$ for the component $T_{zz}$ of the stress tensor and $\Theta$ for its trace. The function $C$ is an explicit linear combination of the two-point functions $\langle T T \rangle$, $\langle T \Theta \rangle$ and $\langle \Theta \Theta \rangle$ at distance $r$. Each of them is multiplied by the power of $z$ and $\bar z$ that makes it dimensionless. Conservation of the stress tensor shows that the derivative of $C$ with respect to $\log r$ is a negative multiple of $r^4 \langle \Theta(r) \Theta(0) \rangle$. Reflection positivity makes this two-point function non-negative, which gives the monotonicity; it vanishes only when $\Theta = 0$, that is, at a conformal point.

Consequence: if a flow starts near a fixed point with central charge $c_{\mathrm{UV}}$ and ends at one with central charge $c_{\mathrm{IR}}$, then $c_{\mathrm{UV}} \ge c_{\mathrm{IR}}$. For example, a massive deformation of the critical Ising model, which has $c = 1/2$, flows to a theory with no long-range degrees of freedom, which has $c = 0$.

Unitarity is necessary for the argument: in non-unitary theories the positivity of $\langle \Theta \Theta \rangle$ fails, and the central charge can increase along a flow.

Going deeper

For a student

The theorem forbids renormalization group flows in two dimensions from being periodic. Since $C$ strictly decreases along nontrivial flow lines, flows run from fixed points with larger $c$ to fixed points with smaller $c$. It is a strong constraint on the phase diagrams of two-dimensional statistical models, whose critical points are described by conformal field theories. In four dimensions the analogous statement for the coefficient $a$ of the trace anomaly is the a-theorem.