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

Abstract

Two strategies are taken into account to determine the $f_1(1420)$-$f_1(1285)$ mixing angle $\theta$. (i) First, using the Gell-Mann-Okubo mass formula together with the $K_1(1270)$-$K_1(1400)$ mixing angle ${\theta }_{K_1}=(-34\pm 13{)}^{\circ }$ 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 $\theta =(23_{-23}^{+17}{)}^{\circ }$. (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 two-fold solution $\theta =(19.4_{-4.6}^{+4.5}{)}^{\circ }$ or $(51.1_{-4.6}^{+4.5}{)}^{\circ }$. Combining these two analyses, we thus obtain $\theta =(19.4_{-4.6}^{+4.5}{)}^{\circ }$. 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 $\mathcal{O}({\alpha }_s^3,m_s^2{\alpha }_s^2)$ accuracy. Using the average of the recent lattice results and the $\theta$ value that we have obtained as inputs, we get $<\bar{s}s>/<\bar{u}u>=0.41\pm 0.09$.

Comments: 10 pages, 1 table, published version