Infrared freezing of Euclidean observables and analyticity in perturbative QCD

The renormalization-group improved finite order expansions of the QCD observables have an unphysical singularity in the Euclidean region, due to the Landau pole of the running coupling. Recently it was claimed that, by using a modified Borel representation, the leading one-chain term in a skeleton expansion of the Euclidean QCD observables is finite and continuous across the Landau pole, and then exhibits an infrared freezing behaviour, vanishing at $Q^2=0$. In the present paper we show, using for illustration the Adler-${\cal D}$ function, that the above Borel prescription violates the causality properties expressed by energy-plane analyticity: the function ${\cal D}(Q^2)$ thus defined is the boundary value of a piecewise analytic function in the complex plane, instead of being a standard analytic function. So, the price to be paid for the infrared freezing of Euclidean QCD observables is the loss of a fundamental property of local quantum field theory.