Renormalization group, causality, and nonpower perturbation expansion in QFT

The structure of the QFT expansion is studied in the framework of a new “invariant analytic” version of the perturbative QCD. Here, an invariant coupling constant $a(Q^2/\Lambda^2)=\beta_1\alpha_s(Q^2)/(4\pi)$ becomes a $Q^2$-analytic invariant function $a_{\mathrm{an}}(Q^2/\Lambda^2)\equiv\mathcal A(x)$, which, by construction, is free of ghost singularities because it incorporates some nonperturbative structures. In the framework of the “analyticized” perturbation theory, an expansion for an observable $F$, instead of powers of the analytic invariant charge $\mathcal A(x)$, may contain specific functions $\mathcal A_n(x)=\left[a^n(x)\right]_{\mathrm{an}}$, the "$n$th power of $a(x)$ analyticized as a whole." Functions $A_{n>2}(x)$ for small $Q^2\leq\Lambda^2$ oscillate, which results in weak loop and scheme dependences. Because of the analyticity requirement, the perturbation series for $F(x)$ becomes an asymptotic expansion á la Erdélyi using a nonpower set $\{\mathcal A_n(x)\}$. The probable ambiguities of the invariant analyticization procedure and the possible inconsistency of some of its versions with the renormalization group structure are also discussed.