Two strategies are taken into account to determine the $f_1(1420)\mathrm{\text{−}}{f}_1(1285)$ mixing angle $\theta$. (i) First, using the Gell-Mann-Okubo mass formula together with the $K_1(1270)\mathrm{\text{−}}{K}_1(1400)$ mixing angle ${\theta }_{K_1}=(-34\pm 13)°$ extracted from the data for $\mathcal{B}(B\to K_1(1270)\gamma )$, $\mathcal{B}(B\to K_1(1400)\gamma )$, $\mathcal{B}(\tau \to K_1(1270){\nu }_{\tau })$, and $\mathcal{B}(\tau \to K_1(1420){\nu }_{\tau })$, gave $\theta =({23}_{-23}^{+17})°$. (ii) Second, from the study of the ratio for $f_1(1285)\to \varphi \gamma$ and $f_1(1285)\to {\rho }^0\gamma$ branching fractions, we have a twofold solution $\theta =({19.4}_{-4.6}^{+4.5})°$ or $({51.1}_{-4.6}^{+4.5})°$. Combining these two analyses, we thus obtain $\theta =({19.4}_{-4.6}^{+4.5})°$. We further compute the strange quark mass and strange quark condensate from the analysis of the $f_1(1420)\mathrm{\text{−}}{f}_1(1285)$ mass difference QCD sum rule, where the operator-product-expansion series is up to dimension six and to $\mathcal{O}({\alpha }_s^3,m_s^2{\alpha }_s^2)$ accuracy. Using the average of the recent lattice results and the $\theta$ value that we have obtained as inputs, we get $⟨\overline{s}s⟩/⟨\overline{u}u⟩=0.41\pm 0.09$.

• Received 30 June 2011

DOI:https://doi.org/10.1103/PhysRevD.84.034035