The $f_{\varrho }/m_{\varrho }$ and $f_{\pi }/m_{\varrho }$ ratios and the conformal window

Abstract

The $f_{\varrho }/m_{\varrho }$ ratio is calculated at N${}^3$LO order within perturbative (p)NRQCD with $N_f$ flavors of mass degenerate fermions. The massless limit of the ratio is expanded á la Banks-Zaks in $ϵ=16.5-N_f$ leading to reliable predictions close to the upper end of the conformal window. The comparison of the NNLO and N${}^3$LO results indicate that the Banks-Zaks expansion may be reliable down to twelve flavors. Previous lattice calculations combined with the KSRF relations provide us with the same ratio for the range $2\le N_f\le 10$. Assuming a monotonous dependence on $N_f$ leads to an estimate for the lower end of the conformal window, $N_f^{\ast }\simeq 12$, by matching the non-perturbative and our perturbative results. In any case an abrupt change is observed in $f_{\varrho }/m_{\varrho }$ at twelve flavors. As a cross-check we also consider the $f_{\pi }/m_{\varrho }$ ratio for which lattice results are also available. The perturbative calculation at present is only at the NNLO level which is insufficient for a reliable and robust matching between the low $N_f$ and high $N_f$ regions. Nonetheless, using the relative size of the N${}^3$LO correction of $f_{\varrho }/m_{\varrho }$ for estimating the same for $f_{\pi }/m_{\varrho }$ leads to the estimate $N_f^{\ast }\simeq 13$.

Comments: 8 pages, 2 figures, published version, references added