Conformal window from conformal expansion

Abstract

We study the conformal window of asymptotically free gauge theories containing $N_f$ flavors of fermion matter transforming to the vector and two-index representations of $SO(N),~SU(N)$ and $Sp(2N)$ gauge groups. For $SO(N)$ we also consider the spinorial representation. We determine the critical number of flavors $N_f^{\rm cr}$, corresponding to the lower end of the conformal window, by using the conjectured critical condition on the anomalous dimension of the fermion bilinear at an infrared fixed point, $\gamma_{\bar{\psi}\psi}=1$ or equivalently $\gamma_{\bar{\psi}\psi}(2-\gamma_{\bar{\psi}\psi})=1$. To compute the anomalous dimension we employ the Banks-Zaks conformal expansion up to the $4$th order in $\Delta_{N_f}=N_f^{\rm AF}-N_f$ with $N_f^{\rm AF}$ denoting the onset of the loss of asymptotic freedom, where we show that the latter critical condition provides a better performance along with this conformal expansion. To quantify the uncertainties in our analysis, which potentially originate from nonperturbative effects, we propose two distinct approaches by assuming the large order behavior of the conformal expansion separately, either convergent or divergent asymptotic. In the former case, we take the difference in the Pad\'e approximants to the two definitions of the critical condition, whereas in the latter case the truncation error associated with the singularity in the Borel plane is taken into account. Our results are further compared to other analytical methods as well as lattice results available in the literature. In particular, we find that $SU(2)$ with six and $SU(3)$ with ten fundamental flavors are likely on the lower edge of the conformal window, which are consistent with the recent lattice results. We also predict that $Sp(4)$ theories with fundamental and antisymmetric fermions have the critical numbers of flavors, approximately ten and five, respectively.