# Irreversibility of the renormalization group flow in non-unitary quantum field theory

## TL;DR

This paper extends the concept of the irreversibility of the renormalization group flow, known from unitary theories, to non-unitary but PT-invariant quantum field theories in two dimensions, showing a monotonic function analogous to the c-theorem.

## Contribution

It generalizes Zamolodchikov's c-theorem to PT-symmetric non-unitary quantum field theories, establishing a monotonic RG flow function under broader conditions.

## Key findings

- Existence of a non-negative, monotonically decreasing c-function along RG flows.
- At critical points, the c-function equals the effective central charge c_eff.
- In rational models, c_eff relates to the central charge and lowest primary field dimension.

## Abstract

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 $c_{\mathrm{eff}}=c-24\Delta$, where $c$ is the central charge and $\Delta$ is the lowest primary field dimension in the conformal field theory which describes the critical point.