Bounds on the Gell-Mann--Low functions in quantum electrodynamics and in the Wess-Zumino model

Abstract

We derive bounds $ |\frac{d\psi(\alpha)}{d\alpha}| \leq 1 $, $ \frac{d(\frac{d\psi(\alpha)}{d\alpha}\psi(\alpha))}{d\alpha} \leq 1 $ on the GL (Gell-Mann--Low) function $\psi(\alpha)$ from the Kallen-Lehmann dispersion representation in quantum electrodynamics. We also derive analogous bounds for the GL function in the Wess-Zumino model. The implications of the obtained inequalities are discussed. In particular, we obtain bounds on coupling constants in dark photon model and in dark matter model with vector $(B-L)$ messenger.