The c_eff-theorem : Irreversibility of RG Flows in 2D Non-unitary QFT’s

We show irreversibility of the renormalization group flow in non-unitary but ${cal PT}$-invariant quantum field theory in two space-time dimensions. In addition to unbroken $mathcal{PT}$-symmetry and a positive energy spectrum, we assume standard properties of quantum field theory including a local energy-momentum tensor and relativistic invariance. This generalizes Zamolodchikov's $c$-theorem to ${cal PT}$-symmetric hamiltonians. Our proof follows closely Zamolodchikov's arguments. We show that a function $c_{mathrm{eff}}(s)$ of the renormalization group parameter $s$ exists which is non-negative and monotonically decreasing along renormalization group flows. Its value at a critical point is the ``effective central charge" entering the specific free energy. At least in rational models, this equals $c_{mathrm{eff}}=c-24Delta$, where $c$ is the central charge and $Delta$ is the lowest primary field dimension in the conformal field theory which describes the critical point.