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The mixing of $D_{s1}(2460)$ and $D_{s1}(2536)$

The mixing mechanism of axial-vectors $D_{s1}(2460)$ and $D_{s1}(2536)$ is studied via intermediate hadron loops, e.g. $D^* K$, to which both states have strong couplings. By constructing the two-state mixing propagator matrix that respects the unitarity constraint and calculating the vertex coupling form factors in a chiral quark model, we can extract the masses, widths and mixing angles of the physical states. Two poles can be identified in the propagator matrix. One is at $\sqrt{s}=2454.5 \ \textrm{MeV}$ corresponding to $D_{s1}(2460)$ and the other at $\sqrt{s}=(2544.9-1.0i) \ \textrm{MeV}$ corresponding to $D_{s1}(2536)$. For $D_{s1}(2460)$, a large mixing angle $θ=47.5^\circ$ between ${}^3P_1$ and ${}^1P_1$ is obtained. It is driven by the real part of the mixing matrix element and corresponds to $θ'=12.3^\circ$ between the $j=1/2$ and $j=3/2$ state mixing in the heavy quark limit. For $D_{s1}(2536)$, a mixing angle $θ=39.7^\circ$ which corresponds to $θ'=4.4^\circ$ in the heavy quark limit is found. An additional phase angle $ϕ=-6.9^\circ \sim 6.9^\circ$ is needed at the pole mass of $D_{s1}(2536)$ since the mixing matrix elements are complex numbers. Both the real and imaginary part are found important for the large mixing angle. We show that the new experimental data from BaBar provide a strong constraint on the mixing angle at the mass of $D_{s1}(2536)$, from which two values can be extracted, i.e. $θ_1=32.1^\circ$ or $θ_2=38.4^\circ$. Our study agrees well with the latter one. Detailed analysis of the mass shift procedure due to the coupled channel effects is also presented.