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$1^{++}$ Nonet Singlet-Octet Mixing Angle, Strange Quark Mass, and Strange Quark Condensate

Two strategies are taken into account to determine the $f_1(1420)$-$f_1(1285)$ mixing angle $θ$. (i) First, using the Gell-Mann-Okubo mass formula together with the $K_1(1270)$-$K_1(1400)$ mixing angle $θ_{K_1}=(-34\pm 13)^\circ$ extracted from the data for ${\cal B}(B\to K_1(1270) γ), {\cal B}(B\to K_1(1400) γ), {\cal B}(τ\to K_1(1270) ν_τ)$, and ${\cal B}(τ\to K_1(1420) ν_τ)$, gave $θ= (23^{+17}_{-23})^\circ$. (ii) Second, from the study of the ratio for $f_1(1285) \to ϕγ$ and $f_1(1285) \to ρ^0γ$ branching fractions, we have two-fold solution $θ=(19.4^{+4.5}_{-4.6})^\circ$ or $(51.1^{+4.5}_{-4.6})^\circ$. Combining these two analyses, we thus obtain $θ=(19.4^{+4.5}_{-4.6})^\circ$. We further compute the strange quark mass and strange quark condensate from the analysis of the $f_1(1420)-f_1(1285)$ mass difference QCD sum rule, where the operator-product-expansion series is up to dimension six and to ${\cal O}(α_s^3, m_s^2 α_s^2)$ accuracy. Using the average of the recent lattice results and the $θ$ value that we have obtained as inputs, we get $<\bar{s} s>/<\bar{u} u> =0.41 \pm 0.09$.