# Where does decay widths in mass mixing matrix come from?

For some time ago I've seen people use complex mass mixing matrix including decay width of the particles. It kind of makes sense, but I could never fully justify it. I would be grateful if you could share some resources or your insights.

Say I have a scalar $$S$$ and term in the lagrangian coupled to b-quarks $$\mathcal{L} \supset \kappa S\bar{b}b$$ This gives rise to mixing between scalar S and QCD bound states $$\bar{b}b$$ having the same quantum numbers say $$\chi_1$$.

The mixing is proportional to $$\kappa$$ as well as to the radial wave function at the origin of $$\chi_1$$. Let's call this mixing $$=\delta m^2$$.

At this stage people state mass mixing matrix to be: $$M^2= \begin{bmatrix} m_P^2-i\Gamma_P m_P & \delta m^2\\ \delta m^2 & m_{\chi_1}^2-i\Gamma_{\chi_1} m_{\chi_1} \end{bmatrix}$$

Usually mass mixing matrix is constructed as taking second order derivatives, but how decay widths end up here?